Polynomials: Definition, Types, and Properties

Algebraic Expressions and Polynomials: Introduction and Definition

Building from Algebraic Expressions

Before defining polynomials, it's helpful to recall what an algebraic expression is. An algebraic expression is a mathematical phrase that combines numbers (constants), variables (symbols representing quantities that can change or are unknown), and the basic arithmetic operations: addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$).

Examples of algebraic expressions include:

• $5x$ (a number multiplied by a variable)
• $y - 3$ (a variable subtracted by a number)
• $2a + 7b$ (sum of terms involving variables)
• $m^2$ (a variable raised to a power)
• $\frac{p}{q}$ (division of variables)
• $\sqrt{z}$ (a variable under a radical sign)
• $w^{-1}$ (a variable with a negative exponent)

Polynomials represent a special and very important category of algebraic expressions. They have a specific structure and follow certain rules regarding the exponents of the variables.

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Formal Definition of a Polynomial

A polynomial is an algebraic expression that consists of one or more terms added together, where each term is a product of a constant (called the coefficient) and one or more variables raised to non-negative integer powers.

Let's break this down and look at the formal definition, starting with polynomials in a single variable.

A polynomial in one variable, say $x$, is an expression of the form:

In this general form, the following conditions must be met:

• $x$ is the variable. A polynomial in one variable contains only one type of variable.
• $n$ is a non-negative integer ($n \ge 0$). This is a crucial part of the definition. The powers (exponents) of the variable must be whole numbers: $0, 1, 2, 3, \ldots$.
• $a_n, a_{n-1}, \ldots, a_2, a_1, a_0$ are coefficients. These are constants that are multiplied by the variable terms. These coefficients can be any real numbers (integers, rational numbers like fractions, or irrational numbers like $\sqrt{2}$). However, in many contexts (like school algebra), coefficients are often restricted to be rational numbers. For the general definition, real numbers are usually assumed.
• Each part like $a_i x^i$ (where $i$ is from $0$ to $n$) is a term of the polynomial.
• The term $a_0 x^0$ simplifies to $a_0 \times 1 = a_0$ (assuming $x \neq 0$; if $x=0$, $0^0$ is often defined as $1$ in this context, or the constant term is simply defined as the term without a variable factor). $a_0$ is called the constant term. It's the term in the polynomial that does not contain the variable $x$.
• If $a_n \neq 0$, then $n$ is the highest power of $x$ present, and it is called the degree of the polynomial.

The definition of a polynomial can be extended to include multiple variables. A term in a polynomial with multiple variables is a product of a constant coefficient and one or more variables, where each variable is raised to a non-negative integer power. A polynomial in multiple variables is a sum of such terms.

Example of a polynomial in multiple variables: $5x^2 y^3 - 2xy + 7z - 4$.

• Term 1: $5x^2 y^3$. Coefficient is $5$. Variable exponents are $2$ (for $x$) and $3$ (for $y$). Both are non-negative integers.
• Term 2: $-2xy$. Coefficient is $-2$. Variable exponents are $1$ (for $x$) and $1$ (for $y$). Both are non-negative integers.
• Term 3: $7z$. Coefficient is $7$. Variable exponent is $1$ (for $z$). It's a non-negative integer.
• Term 4: $-4$. Coefficient is $-4$. This is a constant term, which can be thought of as $-4x^0y^0z^0$. Exponents are $0$, which is a non-negative integer.

Since all terms fit the description (constant $\times$ variables raised to non-negative integer powers), the entire expression is a polynomial.

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Examples of Polynomials and Non-Polynomials

To determine if an algebraic expression is a polynomial, you need to examine each term. The exponents of the variables must be non-negative integers ($0, 1, 2, 3, \ldots$). Expressions involving division by variables, variables under radical signs (unless they simplify to integer exponents), or variables in the exponents are generally NOT polynomials.

Examples of expressions that ARE polynomials:

• $7$: This is a constant term. It can be written as $7x^0$. The exponent is $0$, which is a non-negative integer. This is a polynomial (specifically, a constant polynomial).
• $2x$: The variable $x$ has an exponent of $1$ ($x^1$), which is a non-negative integer. This is a polynomial.
• $y - 10$: This is $y^1 - 10y^0$. The exponents $1$ and $0$ are non-negative integers. This is a polynomial.
• $5x^2 + 3x - 1$: The exponents of $x$ are $2, 1,$ and $0$. All are non-negative integers. The coefficients ($5, 3, -1$) are real numbers. This is a polynomial.
• $m^3 - \sqrt{2} m^2 + \frac{5}{6} m$: The exponents of $m$ are $3, 2,$ and $1$. All are non-negative integers. The coefficients ($1, -\sqrt{2}, \frac{5}{6}$) are real numbers. This is a polynomial.
• $a^2 b^3 + 5ab - c^4 + 1$: This is a polynomial in multiple variables ($a, b, c$). The exponents for $a, b, c$ in each term ($2, 3$; $1, 1$; $4$; $0, 0, 0$ for the constant term) are all non-negative integers.
• $0$: This is the zero polynomial. The exponents are not well-defined, or it can be considered to have any degree less than 0 (often negative infinity). It's considered a polynomial.

Examples of expressions that are NOT polynomials (with reasons):

• $\frac{1}{x}$: This can be written as $x^{-1}$. The exponent is $-1$, which is a negative integer. Therefore, it is not a polynomial.
• $\frac{x+1}{x-2}$: This expression involves division by an expression containing a variable ($x-2$). This is a rational expression, not a polynomial.
• $\sqrt{y}$: This can be written as $y^{1/2}$. The exponent is $\frac{1}{2}$, which is a fraction (not a non-negative integer). Therefore, it is not a polynomial.
• $\sqrt{x+1}$: The variable is under a radical sign, which is equivalent to having a fractional exponent on the entire expression $((x+1)^{1/2})$. Therefore, it is not a polynomial.
• $3^x$: The variable $x$ is in the exponent, not the base. This is an exponential expression, not a polynomial.
• $\frac{5}{a^2}$: This can be written as $5a^{-2}$. The exponent is $-2$, which is a negative integer. Therefore, it is not a polynomial.
• $|x|$: The absolute value function is not a polynomial function.

Understanding the specific requirements for an expression to be a polynomial (non-negative integer exponents on variables, no division by variables, no variables under radicals etc.) is crucial for classifying algebraic expressions and for determining which properties and theorems of polynomials can be applied to them.

Terms Related to Polynomials (Degree, leading coefficient, Monomial, Binomial, Trinomial)

Terminology for Describing Polynomials

To effectively work with and discuss polynomials, it's essential to use precise terminology that describes their structure and characteristics. These terms allow mathematicians to classify polynomials, predict their behavior, and apply appropriate methods for solving related problems (like finding roots or factorising).

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1. Term of a Polynomial

As established in the discussion on algebraic expressions, a term is a single number, a single variable, or the product of numbers and variables raised to non-negative integer powers. In a polynomial, the terms are the individual components that are added or subtracted.

Example: Consider the polynomial $7x^4 - 5x^2 y + 3y - 12$. The terms of this polynomial are $7x^4$, $-5x^2 y$, $3y$, and $-12$. Each of these is a valid monomial that fits the definition of a term in a polynomial.

2. Coefficient of a Term

The coefficient of a term is the numerical factor that multiplies the variable part of that term. It's the constant number associated with the term.

Example: In the polynomial $7x^4 - 5x^2 y + 3y - 12$:

• The coefficient of the term $7x^4$ is $\textbf{7}$.
• The coefficient of the term $-5x^2 y$ is $\textbf{-5}$.
• The coefficient of the term $3y$ is $\textbf{3}$ (since $3y$ is $3 \times y^1$).
• The coefficient of the constant term $-12$ is $\textbf{-12}$.
• If a term is just a variable like $x^2$, its coefficient is implicitly $\textbf{1}$ ($x^2 = 1x^2$).
• If a term is a negative variable like $-y^3$, its coefficient is implicitly $\textbf{-1}$ ($-y^3 = -1y^3$).

3. Degree of a Term

The degree of a term is the sum of the exponents of all the variables present in that specific term. For a constant term (a term without any variables), the degree is considered to be $0$, as it can be thought of as being multiplied by a variable raised to the power of $0$ (e.g., $5 = 5x^0$).

Example: In the polynomial $10x^5 - 4x^2 y^3 + 6x - 1$:

• The term $10x^5$ has only one variable, $x$, with an exponent of $5$. The degree of this term is $\textbf{5}$.
• The term $-4x^2 y^3$ has variables $x$ with exponent $2$ and $y$ with exponent $3$. The degree of this term is the sum of the exponents: $2 + 3 = \textbf{5}$.
• The term $6x$ has one variable, $x$, with an exponent of $1$ ($x = x^1$). The degree of this term is $\textbf{1}$.
• The constant term $-1$ has no variables. Its degree is $\textbf{0}$ (thinking of it as $-1x^0$).

4. Degree of a Polynomial

The degree of a polynomial is the highest degree among all of its terms that have non-zero coefficients. This single number is a fundamental characteristic of the polynomial and significantly influences its properties and the methods used to work with it.

To find the degree of a polynomial:

1. Find the degree of each individual term (sum of variable exponents).
2. The highest of these term degrees is the degree of the polynomial. (Ignore terms with a coefficient of zero, as they don't affect the degree).

Example: Consider the polynomial $10x^5 - 4x^2 y^3 + 6x - 1$. We found the degrees of the terms to be $5, 5, 1, 0$. The highest degree among these is $5$. Therefore, the degree of the polynomial is $\textbf{5}$.

Example: Consider the polynomial $P(x) = 7x^2 - 3x^5 + 2x^3 + 9$. The terms are $7x^2, -3x^5, 2x^3, 9$. Their degrees are $2, 5, 3, 0$, respectively. The highest degree is $5$. The degree of the polynomial $P(x)$ is $\textbf{5}$. When written in standard form (decreasing powers of $x$), $P(x) = -3x^5 + 2x^3 + 7x^2 + 9$, the highest degree term is clearly visible.

Example: A constant polynomial $c$, where $c \neq 0$ (e.g., $25$). This can be written as $25x^0$. The only term has degree $0$. The degree of a non-zero constant polynomial is $\textbf{0}$.

Example: The zero polynomial, $P(x) = 0$. This polynomial has no non-zero terms. Its degree is a special case and is usually considered to be undefined. Sometimes, for certain applications, it might be defined as $-\infty$, but "undefined" is the most common convention.

5. Classification by Degree

Polynomials are frequently categorized based on their degree, as the degree often indicates the number of roots (solutions) a polynomial equation might have and influences the shape of its graph. Common classifications include:

• Constant Polynomial: A polynomial of degree $0$.
  Examples: $15, -7, \sqrt{5}$. (Any non-zero constant).
• Linear Polynomial: A polynomial of degree $1$. The general form in one variable is $ax+b$, where $a \neq 0$.
  Examples: $2x + 3, y - 8, 5p$.
• Quadratic Polynomial: A polynomial of degree $2$. The general form in one variable is $ax^2+bx+c$, where $a \neq 0$.
  Examples: $x^2 - 5x + 6, 3y^2 + 4, 2m^2$.
• Cubic Polynomial: A polynomial of degree $3$. The general form in one variable is $ax^3+bx^2+cx+d$, where $a \neq 0$.
  Examples: $x^3 - 1, 2y^3 + 5y^2 - y + 7$.
• Quartic Polynomial: A polynomial of degree $4$.
• Quintic Polynomial: A polynomial of degree $5$.
• And so on for higher degrees.

6. Leading Term and Leading Coefficient

For a polynomial in a single variable, when the terms are written in order of decreasing powers of the variable (this is called the standard form of the polynomial), the term with the highest degree is called the leading term. The coefficient of this leading term is called the leading coefficient.

The leading term and leading coefficient are particularly important as they determine the polynomial's behavior for very large positive or negative values of the variable (the end behavior of the graph).

Example: Consider the polynomial $P(x) = 8x^3 - 5x^7 + 2x - 1$. First, write it in standard form (decreasing powers of $x$):

$$ P(x) = -5x^7 + 8x^3 + 2x - 1 $$

The highest degree term is $-5x^7$. The leading term is $\textbf{-5x\textsuperscript{7}}$. The coefficient of this term is $-5$. So, the leading coefficient is $\textbf{-5}$. The degree of this polynomial is $7$.

Example: Consider the polynomial $Q(y) = y^2 - 5y + 1$. It is already in standard form. The highest degree term is $y^2$. The leading term is $\textbf{y\textsuperscript{2}}$. The coefficient of $y^2$ is $1$. So, the leading coefficient is $\textbf{1}$. The degree of this polynomial is $2$.

7. Classification by Number of Terms

Polynomials are also classified based on how many terms they contain after all like terms have been combined and the expression is simplified. This classification is simpler than by degree but also useful for identification.

• Monomial: A polynomial consisting of exactly one term.
  Examples: $5x^3$, $-7y$, $10$, $a^2 b^3$, $-p$.
• Binomial: A polynomial consisting of exactly two terms connected by addition or subtraction.
  Examples: $x + y$, $a^2 - 4$, $2p + 3q$, $m^4 - n^4$.
• Trinomial: A polynomial consisting of exactly three terms connected by addition or subtraction.
  Examples: $x^2 + 2x + 1$, $a^2 - ab + b^2$, $4m^2 - 3m + 5$.
• For polynomials with more than three terms, there isn't a specific common name (like "quadrinomial"). They are simply referred to as polynomials. The term "polynomial" itself means "many terms" (poly- meaning many, -nomial meaning term). Thus, technically, monomials, binomials, and trinomials are all types of polynomials.

Mastering these terms – term, coefficient, degree of a term, degree of a polynomial, leading term, leading coefficient, and the classification by the number of terms – provides the essential vocabulary for understanding and working with polynomials in algebra.

Value of a Polynomial

Evaluating a Polynomial at a Specific Point

Just like a general algebraic expression, the numerical value of a polynomial is not fixed. It changes depending on the specific numbers assigned to the variables within it. The process of finding this numerical value is called evaluating the polynomial. The resulting number is known as the value of the polynomial for those particular variable assignments.

If we have a polynomial denoted by $P(x)$, where $x$ is the variable, and we want to find its value when $x$ takes on a specific number, say $c$, we write this as $P(c)$. To calculate $P(c)$, we substitute the number $c$ for every occurrence of the variable $x$ in the polynomial expression and then carry out all the indicated arithmetic operations according to the standard order.

For a polynomial in multiple variables, say $Q(x, y)$, evaluating it at specific values, for example, $x=a$ and $y=b$, is denoted as $Q(a, b)$. We substitute $a$ for every $x$ and $b$ for every $y$ and calculate the result.

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Procedure for Evaluating a Polynomial

To find the value of a polynomial for given numerical values of its variables, follow these steps:

1. Substitution:

   Replace each variable in the polynomial with its corresponding given numerical value. It is a good practice to enclose the substituted number within parentheses, especially when dealing with negative numbers, fractions, or when the variable is raised to a power or multiplied by a coefficient. This helps maintain the correct signs and ensures operations are performed correctly.

2. Calculation:

   Compute the value of the resulting numerical expression using the correct order of operations (BODMAS / PEMDAS).
   • Brackets / Parentheses: Evaluate expressions within grouping symbols first.
   • Orders / Exponents: Calculate powers and roots.
   • Division and Multiplication: Perform these from left to right.
   • Addition and Subtraction: Perform these from left to right.

Polynomials with real coefficients are defined for all real numbers. Therefore, you can substitute any real number for the variables, and the resulting arithmetic expression will always be defined.

Answer:

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Answer:

Evaluating polynomials is a fundamental operation used in various contexts, such as finding points on the graph of a polynomial function, checking solutions to polynomial equations, or calculating specific outcomes in applied problems modelled by polynomials.

Zero of a Polynomial

Defining the Zeros (Roots)

A key concept in the study of polynomials is the idea of a "zero". A number is called a zero of a polynomial if, when that number is substituted for the variable in the polynomial expression, the value of the polynomial becomes $0$.

More formally, for a polynomial $P(x)$ in one variable $x$, a number $c$ is a zero of $P(x)$ if and only if evaluating the polynomial at $x=c$ results in $0$. That is:

The terms zero of a polynomial and root of a polynomial are often used interchangeably. The term "root" is more commonly used when the polynomial is set equal to zero to form an equation. Thus, the zeros of the polynomial $P(x)$ are the solutions or roots of the polynomial equation $P(x) = 0$. Finding these zeros or roots is a central problem in algebra with wide applications.

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Methods for Finding Zeros (Simple Cases)

Finding the zeros of a polynomial means determining the specific value(s) of the variable(s) that make the polynomial expression evaluate to zero. The general strategy is to set the polynomial expression equal to zero and solve the resulting equation. The complexity of finding the zeros depends heavily on the degree of the polynomial.

1. Zero of a Constant Polynomial

A non-zero constant polynomial has the form $P(x) = c$, where $c$ is a specific constant number, and $c \neq 0$. To find its zeros, we set it equal to zero:

$$ P(x) = 0 \implies c = 0 $$

Since we assumed $c \neq 0$, this equation ($c=0$) is a contradiction. It has no solution for $x$. Therefore, a non-zero constant polynomial (like $P(x)=5$ or $P(x)=-2$) has no zeros.

The only constant polynomial that has zeros is the zero polynomial, $P(x) = 0$. When we set $P(x) = 0$, we get $0 = 0$, which is true for any value of $x$. Therefore, every real number is a zero of the zero polynomial.

2. Zero of a Linear Polynomial

A linear polynomial in one variable has the general form $P(x) = ax + b$, where $a$ and $b$ are constants, and crucially, $a \neq 0$. To find its zero, we set the polynomial equal to zero and solve the resulting linear equation:

$$ P(x) = 0 $$ $$ ax + b = 0 $$

Subtract $b$ from both sides:

$$ ax = -b $$

Divide by $a$ (which is non-zero):

A linear polynomial with a non-zero leading coefficient ($a \neq 0$) always has exactly one unique real zero, given by the formula $x = -\frac{b}{a}$.

Answer:

3. Zeros of a Quadratic Polynomial

A quadratic polynomial in one variable has the general form $P(x) = ax^2 + bx + c$, where $a, b, c$ are constants and $a \neq 0$. Finding its zeros involves solving the quadratic equation $ax^2 + bx + c = 0$. Quadratic equations can have different types and numbers of solutions (zeros):

• Two distinct real zeros: If the discriminant, $b^2 - 4ac$, is positive ($b^2 - 4ac > 0$).
• Exactly one real zero (a repeated or double root): If the discriminant is zero ($b^2 - 4ac = 0$).
• No real zeros (two complex conjugate zeros): If the discriminant is negative ($b^2 - 4ac < 0$). (Complex numbers are introduced later in the notes).

There are several methods to find the zeros of a quadratic polynomial:

• Factoring: If the quadratic expression $ax^2 + bx + c$ can be factored into a product of two linear factors, say $(px + q)(rx + s)$, then the zeros are found by setting each factor equal to zero and solving for $x$. For example, if it factors into $a(x - \alpha)(x - \beta) = 0$, the zeros are $x=\alpha$ and $x=\beta$. This is often the quickest method when factorisation is straightforward.
• Using the Quadratic Formula: This formula can be used to find the roots of any quadratic equation, regardless of whether it is easily factorable. The zeros of $ax^2 + bx + c = 0$ are given by:

  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

  [Quadratic Formula]

  The term $b^2 - 4ac$ inside the square root is the discriminant, which determines the nature of the roots.

• Completing the Square: This method involves manipulating the quadratic equation algebraically to transform one side into a perfect square trinomial, resulting in an equation of the form $(x+h)^2 = k$, which can then be solved by taking the square root of both sides ($\pm \sqrt{k}$).

Answer:

Number of Zeros

The number of zeros a polynomial has is related to its degree. A fundamental result in algebra, the Fundamental Theorem of Algebra, states that a polynomial of degree $n \ge 1$ with complex coefficients has exactly $n$ complex roots (zeros), when counted with multiplicity (meaning if a root appears multiple times in factoring, it is counted that many times).

When we are specifically looking for real zeros of a polynomial with real coefficients:

• A constant polynomial $P(x) = c$: Has no real zeros if $c \neq 0$. Every real number is a zero if $c = 0$.
• A linear polynomial (degree 1): Has exactly one real zero (unless it's the zero polynomial).
• A quadratic polynomial (degree 2): Has at most two real zeros. (It can have two distinct real zeros, one repeated real zero, or no real zeros if its roots are complex).
• A cubic polynomial (degree 3): Has at most three real zeros.
• In general, a polynomial of degree $n$ with real coefficients has at most $n$ distinct real zeros. However, it will always have exactly $n$ complex zeros (counting multiplicity).

Finding the zeros of polynomials of degree 3 or higher can be more challenging and often involves techniques like the Rational Root Theorem, synthetic division (using potential rational roots to find factors), or numerical methods when exact analytical solutions are difficult or impossible to find.

Geometrical Representation of a Polynomial (Graphing Zeroes)

Connecting Algebra and Geometry

Mathematics often benefits from visual representations. Algebraic expressions, especially polynomials, can be represented geometrically using graphs. For a polynomial in one variable, say $P(x)$, we can associate it with a function $y = P(x)$. By assigning various values to $x$, calculating the corresponding value of $y = P(x)$, and plotting these $(x, y)$ pairs on a coordinate plane, we can sketch the graph of the polynomial. This graph is typically a smooth, continuous curve.

The graph of a polynomial provides a powerful visual tool for understanding its properties. In particular, there is a direct and important connection between the zeros of a polynomial and its graph.

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Geometrical Meaning of Zeros

Recall that the zeros of a polynomial $P(x)$ are the values of $x$ for which $P(x) = 0$. When we plot the graph of $y = P(x)$, the y-coordinate of any point on the graph is given by the value of the polynomial $P(x)$ for the corresponding x-coordinate. The points on the coordinate plane where the y-coordinate is zero are the points that lie on the x-axis.

Therefore, the real zeros of a polynomial $P(x)$ are the x-coordinates of the points where the graph of $y = P(x)$ intersects or touches the x-axis. These points are also known as the x-intercepts of the graph.

• If the graph crosses the x-axis at a point, it indicates that the polynomial changes sign at that x-value. This point corresponds to a real zero with an odd multiplicity (e.g., a single root, a triple root, etc.).
• If the graph touches the x-axis at a point but does not cross it, it means the polynomial's value is zero at that point but doesn't change sign immediately around it. This point corresponds to a real zero with an even multiplicity (e.g., a double root, a quadruple root, etc.). The vertex of a parabola touching the x-axis is a common example of this.

The number of distinct real zeros of a polynomial corresponds to the number of distinct points where its graph intersects or touches the x-axis. A polynomial of degree $n$ can have at most $n$ distinct real zeros, and its graph can intersect or touch the x-axis at most $n$ times.

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Visualizing Zeros Through Graphs

Let's examine the graphs of different degrees of polynomials to see how their real zeros are represented geometrically.

1. Graph of a Linear Polynomial ($y = ax + b$, $a \neq 0$)

The graph of a linear polynomial is always a straight line. As long as the leading coefficient $a$ is not zero, the line is not horizontal and must intersect the x-axis at exactly one point.

The zero of the linear polynomial $P(x) = ax + b$ is found by solving $ax+b=0$, which gives $x = -b/a$. On the graph of $y = ax + b$, this value $x = -b/a$ is the x-coordinate of the single point where the line crosses the x-axis (the x-intercept).

The image shows a straight line graph. The point where it crosses the x-axis (the x-intercept) is the location of the polynomial's zero. The x-coordinate of this point is the value of the zero.

2. Graph of a Quadratic Polynomial ($y = ax^2 + bx + c$, $a \neq 0$)

The graph of a quadratic polynomial is a parabola. A parabola can intersect or touch the x-axis in three possible ways, corresponding to the number of distinct real zeros:

• Two distinct real zeros: The parabola crosses the x-axis at two different points. The x-coordinates of these two points are the two distinct real zeros of the quadratic polynomial. This occurs when the discriminant ($b^2 - 4ac$) is positive.

  Example: $P(x) = x^2 - 4$. The zeros are $x=2$ and $x=-2$. The graph of $y = x^2 - 4$ is a parabola opening upwards, and it intersects the x-axis at the points $(2, 0)$ and $(-2, 0)$.

  

  The image shows a parabola crossing the x-axis at two different points. The x-coordinates of these intersection points are the two real zeros.

• Exactly one real zero (a repeated or double root): The parabola touches the x-axis at exactly one point. This point is the vertex of the parabola, and it lies directly on the x-axis. The x-coordinate of this single point is the value of the repeated real zero. This occurs when the discriminant ($b^2 - 4ac$) is zero.

  Example: $P(x) = x^2 - 4x + 4$. This can be factored as $P(x) = (x-2)^2$. The zero is $x=2$ with multiplicity 2 (a double root). The graph of $y = (x-2)^2$ touches the x-axis at the point $(2, 0)$.

  

  The image shows a parabola touching the x-axis at one point. The x-coordinate of this tangent point is the single real zero.

• No real zeros: The parabola does not intersect or touch the x-axis at all. It lies entirely above the x-axis (if $a>0$) or entirely below the x-axis (if $a<0$). In this case, the quadratic polynomial has two complex conjugate zeros but no real zeros. This occurs when the discriminant ($b^2 - 4ac$) is negative.

  Example: $P(x) = x^2 + 1$. The zeros are $x = \pm i$ (imaginary numbers). The graph of $y = x^2 + 1$ is a parabola shifted upwards by 1 unit and does not intersect the x-axis.

  

  The image shows a parabola that does not touch or cross the x-axis, indicating there are no real zeros.

3. Graph of a Cubic Polynomial ($y = ax^3 + bx^2 + cx + d$, $a \neq 0$)

The graph of a cubic polynomial is a more complex curve than a parabola. A cubic polynomial (degree 3) must cross the x-axis at least once (because odd-degree polynomials with real coefficients always have at least one real root). It can intersect or touch the x-axis at one, two, or three distinct points. Thus, a cubic polynomial can have one, two, or three distinct real zeros.

• One real zero: The graph crosses the x-axis at only one point. The other two roots are complex conjugates.
• Two real zeros: The graph crosses the x-axis at one point and touches the x-axis at another point. The point where it touches corresponds to a repeated real zero (e.g., multiplicity 2). The other zero is a distinct real zero.
• Three real zeros: The graph crosses the x-axis at three distinct points. Each of these points corresponds to a distinct real zero.

The image shows examples of cubic graphs demonstrating these possibilities: a graph crossing once (one real zero), a graph touching and crossing (two distinct real zeros, one with multiplicity 2), and a graph crossing three times (three distinct real zeros).

4. General Polynomials

For a general polynomial $P(x)$ of degree $n$, the graph of $y = P(x)$ can have at most $n$ x-intercepts. The x-coordinates of these intercepts are the real zeros of the polynomial. The behavior of the graph (crossing vs. touching) at each x-intercept indicates the multiplicity of the corresponding zero.

Visualizing the graph of a polynomial is a powerful way to understand the concept of zeros and their relationship to the polynomial equation $P(x) = 0$. The graph provides a visual summary of where the polynomial's value is zero.

Relationship Between Zeroes and Coefficients of a Polynomial

One of the most elegant and useful concepts in polynomial algebra is the direct connection between the roots (zeros) of a polynomial and its coefficients. These connections are formally described by Vieta's formulas, named after the French mathematician François Viète.

The basis for these relationships lies in the Factor Theorem, which states that if $c$ is a zero of a polynomial $P(x)$, then $(x-c)$ is a factor of $P(x)$. If a polynomial of degree $n$ has zeros $\alpha_1, \alpha_2, \ldots, \alpha_n$, then it can be written in its factored form as $P(x) = a_n(x-\alpha_1)(x-\alpha_2)\ldots(x-\alpha_n)$, where $a_n$ is the leading coefficient. By expanding this factored form and comparing the coefficients of the corresponding powers of $x$ with the standard form of the polynomial, we can derive these relationships.

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Relationship for a Linear Polynomial (Degree 1)

Consider a linear polynomial in one variable, $P(x) = ax + b$, where $a$ and $b$ are real coefficients and $a \neq 0$.

A linear polynomial of degree 1 has exactly one complex root, which in this case is always a real root. Let this unique zero be $\alpha$. By definition, $P(\alpha) = 0$.

Setting the polynomial equal to zero to find the root:

$$ ax + b = 0 $$

Subtract $b$ from both sides:

$$ ax = -b $$

Divide both sides by $a$ (since $a \neq 0$):

$$ x = -\frac{b}{a} $$

So, the unique zero is $\alpha = -\frac{b}{a}$.

This gives the relationship between the single zero and the coefficients of a linear polynomial:

For a linear polynomial, the sum of the zeros is just the zero itself, and the product of the zeros is also just the zero itself (this relationship becomes more meaningful for polynomials of degree 2 or higher).

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Relationship for a Quadratic Polynomial (Degree 2)

Consider a quadratic polynomial in one variable, $P(x) = ax^2 + bx + c$, where $a, b, c$ are real coefficients and $a \neq 0$. A quadratic polynomial of degree 2 has exactly two complex roots (zeros), counting multiplicity. Let the two zeros be denoted by $\alpha$ and $\beta$. These roots can be real or complex.

If $\alpha$ and $\beta$ are the zeros of $P(x)$, then $(x - \alpha)$ and $(x - \beta)$ are factors of $P(x)$. Therefore, $P(x)$ can be written in factored form as $P(x) = k(x - \alpha)(x - \beta)$ for some constant $k$.

The leading term of $P(x) = ax^2 + bx + c$ is $ax^2$, with coefficient $a$.

The leading term of $k(x - \alpha)(x - \beta)$ is $k \times x \times x = kx^2$, with coefficient $k$.

Comparing the leading coefficients, we find that $k = a$.

So, we have the identity:

$$ ax^2 + bx + c = a(x - \alpha)(x - \beta) $$

Now, let's expand the right side of this equation:

$$ a(x - \alpha)(x - \beta) = a(x \times x - x \times \beta - \alpha \times x + \alpha \times \beta) $$ $$ = a(x^2 - \beta x - \alpha x + \alpha \beta) $$

Group the terms containing $x$:

$$ = a(x^2 - (\alpha + \beta)x + \alpha \beta) $$

Distribute the constant $a$ into the terms inside the parentheses:

$$ = ax^2 - a(\alpha + \beta)x + a(\alpha \beta) $$

Now, we equate the coefficients of corresponding powers of $x$ from the standard form $ax^2 + bx + c$ and the expanded factored form $ax^2 - a(\alpha + \beta)x + a(\alpha \beta)$:

• Comparing the coefficients of $x^2$: $a = a$. This is consistent.
• Comparing the coefficients of $x$: $b = -a(\alpha + \beta)$.

  To find the sum of the roots, divide both sides by $-a$ (since $a \neq 0$):

  $\alpha + \beta = -\frac{b}{a}$

  [Sum of Zeros of Quadratic]

  This means the sum of the zeros is the negative of the ratio of the coefficient of $x$ to the coefficient of $x^2$.

• Comparing the constant terms: $c = a(\alpha \beta)$.

  To find the product of the roots, divide both sides by $a$:

  $\alpha \beta = \frac{c}{a}$

  [Product of Zeros of Quadratic]

  This means the product of the zeros is the ratio of the constant term to the coefficient of $x^2$.

These two relationships for quadratic polynomials are fundamental Vieta's formulas for degree 2.

Answer:

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Relationship for a Cubic Polynomial (Degree 3)

Consider a cubic polynomial in one variable, $P(x) = ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are real coefficients and $a \neq 0$. A cubic polynomial of degree 3 has exactly three complex roots, counting multiplicity. Let these three zeros be denoted by $\alpha, \beta,$ and $\gamma$.

Similar to the quadratic case, if $\alpha, \beta, \gamma$ are the zeros, then $(x-\alpha)$, $(x-\beta)$, and $(x-\gamma)$ are factors of $P(x)$. Thus, $P(x) = a(x-\alpha)(x-\beta)(x-\gamma)$ since $a$ is the leading coefficient.

Let's expand the right side:

$$ a(x-\alpha)(x-\beta)(x-\gamma) = a(x-\alpha)(x^2 - \gamma x - \beta x + \beta \gamma) $$ $$ = a(x-\alpha)(x^2 - (\beta + \gamma)x + \beta \gamma) $$

Now, distribute $x$ and $-\alpha$ from the first factor:

$$ = a [ x(x^2 - (\beta + \gamma)x + \beta \gamma) - \alpha(x^2 - (\beta + \gamma)x + \beta \gamma) ] $$ $$ = a [ (x^3 - (\beta + \gamma)x^2 + \beta \gamma x) - (\alpha x^2 - \alpha(\beta + \gamma)x + \alpha\beta\gamma) ] $$

Distribute the negative sign in the second parenthesis and $\alpha$:

$$ = a [ x^3 - (\beta + \gamma)x^2 + \beta \gamma x - \alpha x^2 + \alpha\beta x + \alpha\gamma x - \alpha\beta\gamma ] $$

Group terms by powers of $x$ ($x^3$, $x^2$, $x$, constant):

$$ = a [ x^3 + (-\beta - \gamma - \alpha)x^2 + (\beta \gamma + \alpha\beta + \alpha\gamma)x - \alpha\beta\gamma ] $$

Rewrite the coefficients and distribute the outer $a$:

$$ = a x^3 - a(\alpha + \beta + \gamma)x^2 + a(\alpha\beta + \beta\gamma + \gamma\alpha)x - a(\alpha\beta\gamma) $$

Comparing this to the standard form $ax^3 + bx^2 + cx + d$, we equate the coefficients of corresponding powers of $x$:

• Coefficient of $x^3$: $a = a$. (Consistent).
• Coefficient of $x^2$: $b = -a(\alpha + \beta + \gamma)$. Dividing by $-a$:

  $\alpha + \beta + \gamma = -\frac{b}{a}$

  [Sum of Zeros of Cubic]

  The sum of the zeros is the negative of the ratio of the coefficient of $x^2$ to the coefficient of $x^3$.

• Coefficient of $x$: $c = a(\alpha \beta + \beta \gamma + \gamma \alpha)$. Dividing by $a$:

  $\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}$

  [Sum of Pairwise Products]

  The sum of the products of the zeros taken two at a time is the ratio of the coefficient of $x$ to the coefficient of $x^3$.

• Constant term: $d = -a(\alpha \beta \gamma)$. Dividing by $-a$:

  $\alpha \beta \gamma = -\frac{d}{a}$

  [Product of Zeros of Cubic]

  The product of the zeros is the negative of the ratio of the constant term to the coefficient of $x^3$.

These are the fundamental Vieta's formulas for a cubic polynomial.

Answer:

Answer:

Generalization: Vieta's Formulas

The relationships observed for quadratic and cubic polynomials are specific cases of a general set of formulas known as Vieta's formulas, which apply to polynomials of any degree.

For a general polynomial of degree $n$ in one variable $x$, written in standard form:

$$ P(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_1 x^1 + a_0 $$

where $a_n \neq 0$, and its $n$ complex roots (zeros) are $\alpha_1, \alpha_2, \ldots, \alpha_n$ (counting multiplicity), Vieta's formulas describe the sums and products of these roots in terms of the coefficients:

• Sum of the roots: The sum of the roots is the negative of the ratio of the coefficient of $x^{n-1}$ to the coefficient of $x^n$. $$ \sum_{i=1}^n \alpha_i = \alpha_1 + \alpha_2 + \ldots + \alpha_n = -\frac{a_{n-1}}{a_n} $$
• Sum of the products of roots taken two at a time: $$ \sum_{1 \le i < j \le n} \alpha_i \alpha_j = \alpha_1\alpha_2 + \alpha_1\alpha_3 + \ldots + \alpha_{n-1}\alpha_n = \frac{a_{n-2}}{a_n} $$
• Sum of the products of roots taken three at a time: $$ \sum_{1 \le i < j < k \le n} \alpha_i \alpha_j \alpha_k = \alpha_1\alpha_2\alpha_3 + \ldots = -\frac{a_{n-3}}{a_n} $$
• This pattern continues, with the signs alternating.
• The product of all roots: $$ \alpha_1 \alpha_2 \ldots \alpha_n = (-1)^n \frac{a_0}{a_n} $$

  The sign is positive if $n$ is even and negative if $n$ is odd.

Vieta's formulas are incredibly useful for checking polynomial roots, finding roots when some are already known, constructing polynomials with specified roots, and proving various theorems about polynomials.