Chapter 2 Polynomials (Concepts)

Basic Terms Related to Polynomials

In algebra, we work with expressions formed by combining variables and constants using operations like addition, subtraction, multiplication, and division. Examples include $x+2$, $3y^2-5$, $\frac{z}{4}+1$, etc. A polynomial is a specific and very important type of algebraic expression that adheres to certain rules regarding the powers of the variables involved.

What is a Polynomial?

The word "Polynomial" is derived from two words: "Poly" meaning many, and "nomial" meaning term. So, a polynomial is an algebraic expression that consists of many terms. However, there are specific rules for an expression to be a polynomial.

A polynomial in one variable (let's say $x$) is an algebraic expression where the terms are formed by a constant (called a coefficient) multiplied by the variable raised to a non-negative integer power. Non-negative integers are the set of whole numbers: $\{0, 1, 2, 3, ...\}$.

General Form of a Polynomial

The standard way to write a polynomial in the variable $x$ is:

$$p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0$$

Let's break down each part of this form:

• $p(x)$: This is just a way to name the polynomial. It's read as "p of x," meaning a polynomial named 'p' in terms of the variable 'x'.
• $x$: This is the variable. Its value can change.
• $a_n, a_{n-1}, ..., a_1, a_0$: These are the coefficients. They are constant real numbers (which can be integers, fractions, or irrational numbers like $\sqrt{2}$). The coefficient $a_n$ of the highest power of $x$ is called the leading coefficient, provided $a_n \neq 0$.
• $n, n-1, ..., 2, 1, 0$: These are the exponents (or powers) of the variable. The key rule for a polynomial is that these exponents must be whole numbers (0, 1, 2, ...).
• Terms: Each part of the polynomial separated by a '+' or '-' sign is a term. For example, in $2x^2 - 5x + 3$, the terms are $2x^2$, $-5x$, and $3$.

The Constant Term

The constant term in a polynomial is the term that does not contain a variable. It's a standalone number.

In the general form $p(x) = a_n x^n + ... + a_1 x + a_0$, the constant term is $a_0$.

This is because it can be thought of as $a_0 x^0$. Since any non-zero number raised to the power of 0 is 1 ($x^0 = 1$), the term is simply $a_0 \times 1 = a_0$.

Examples:

• In the polynomial $3x^2 + 5x - 7$, the constant term is -7.
• In the polynomial $y^5 - \frac{1}{3}y^2 + \sqrt{5}$, the constant term is $\sqrt{5}$.
• In the polynomial $4z^3 - 2z$, the constant term is 0 (since it can be written as $4z^3 - 2z + 0$).

Identifying Polynomials: Examples

The most important rule to remember is that the exponents of the variable must be whole numbers (0, 1, 2, ...). An expression is NOT a polynomial if:

• The variable is in the denominator (e.g., $\frac{1}{x}$, which is $x^{-1}$).
• The variable is under a radical sign (e.g., $\sqrt{x}$, which is $x^{1/2}$).
• The variable has a fractional or negative exponent.

The following table clarifies this with examples:

Degree of a Polynomial

The degree of a polynomial in one variable is the highest power of the variable in the polynomial, provided that the coefficient of the term with the highest power is not zero.

• For the polynomial $p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, if $a_n \neq 0$, the degree is $n$.
• The degree of a non-zero constant polynomial (e.g., 7, -3.5) is 0, because any non-zero constant $c$ can be written as $c x^0$.
• The zero polynomial is the polynomial where all coefficients are zero, i.e., $p(x) = 0$. The degree of the zero polynomial is generally considered undefined, or sometimes taken as $-\infty$ in advanced contexts.

Classification of Polynomials

Polynomials can be classified into different types based on three main criteria: the number of variables they contain, their degree, and the number of terms they have.

1. Classification Based on Number of Variables

This is the most fundamental classification, distinguishing polynomials by how many different variables they involve.

Polynomial in One Variable

A polynomial that contains only one type of variable (like $x$ or $y$ or $z$, but not a mix). Most of the polynomials studied in early algebra are of this type.

Examples:

• $p(x) = 3x^2 + 5x - 2$ (only contains the variable $x$)
• $q(y) = y^3 - 8$ (only contains the variable $y$)

Polynomial in Two or More Variables

A polynomial that contains two or more different variables.

Examples:

• $p(x, y) = x^2 + 2xy + y^2$ (contains variables $x$ and $y$)
• $q(a, b, c) = a + b + c$ (contains variables $a$, $b$, and $c$)

Note on Degree: For polynomials with more than one variable, the degree of a term is the sum of the exponents of the variables in that term. The degree of the entire polynomial is the highest degree of any of its terms. For example, in $7x^3y^2 + 2xy - 5$, the degree of the first term is $3+2=5$, so the degree of the polynomial is 5.

2. Classification Based on Degree

The degree of a polynomial in one variable is the highest exponent of the variable in any of its terms. This is one of the most common ways to classify polynomials.

Polynomials of degree 5 are called quintic, degree 6 are sextic, and so on, but these names are less commonly used.

3. Classification Based on Number of Terms

Polynomials are also named based on how many non-zero terms they contain. This classification can apply to polynomials of any degree or number of variables.

Monomial

A polynomial with exactly one term. (Mono- means one).

• Examples: $7x^5$, $-4$, $12y^2$, $6x^2y$

Binomial

A polynomial with exactly two terms. (Bi- means two).

• Examples: $x+y$, $3m^4 - 5$, $z^2 + 1$

Trinomial

A polynomial with exactly three terms. (Tri- means three).

• Examples: $x^2 + 5x + 6$, $a^3 - 2a^2 + 7a$

An expression with more than three terms is usually just called a polynomial rather than having a specific name.

Putting It All Together: Combined Classification

The three classification systems (by variables, degree, and terms) are independent of each other. This means any single polynomial fits into one category from each system. To fully describe a polynomial, you can combine these classifications.

Here are some examples showing how one polynomial can have multiple classifications:

Equal Polynomials

Two polynomials are said to be equal if they are identical in their structure. This means they must satisfy two conditions:

• They must have the same degree.
• The coefficients of their corresponding terms (like powers of the variable) must be equal.

Sometimes, a polynomial may need to be simplified or expanded to its standard form before it can be compared with another.

Condition for Equality

Let's consider two polynomials, $p(x)$ and $q(x)$:

$p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$

$q(x) = b_n x^n + b_{n-1} x^{n-1} + ... + b_1 x + b_0$

Then, $p(x) = q(x)$ if and only if $a_i = b_i$ for all $i$ from 0 to $n$. In simpler terms, $a_n = b_n$, $a_{n-1} = b_{n-1}$, and so on, down to $a_0 = b_0$.

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Value and Zero of a Polynomial

A polynomial is a function of its variable(s). Just like other functions, we can evaluate a polynomial for specific numerical values of the variable(s). When the value of the polynomial becomes zero for a particular input value of the variable, that input value holds special significance and is called a 'zero' of the polynomial.

Value of a Polynomial

The value of a polynomial $p(x)$ at a specific real number, say $x = a$, is the result obtained when we substitute the value $a$ for the variable $x$ throughout the polynomial expression. This value is denoted by $p(a)$.

To find the value of $p(x)$ at $x=a$, simply replace every instance of $x$ in the polynomial expression with the number $a$ and then simplify the resulting numerical expression.

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Zero of a Polynomial

A zero of a polynomial $p(x)$ is a specific real number $a$ such that when substituted for $x$, the value of the polynomial becomes 0. In other words, $a$ is a zero of $p(x)$ if $p(a) = 0$.

The zeros of a polynomial are also referred to as the roots of the polynomial equation $p(x) = 0$. Finding the zeros of a polynomial is a fundamental problem in algebra.

The number of zeros a polynomial can have is related to its degree:

• A non-zero constant polynomial (degree 0), like $p(x) = 5$, has no zeros, because $p(x)$ is never equal to 0.
• A linear polynomial (degree 1), of the form $p(x) = ax + b$ where $a \neq 0$, has exactly one zero. This zero can be found by setting $p(x) = 0$: $ax + b = 0 \implies ax = -b \implies x = -\frac{b}{a}$.
• A quadratic polynomial (degree 2), of the form $p(x) = ax^2 + bx + c$ where $a \neq 0$, has at most two zeros. It can have two distinct real zeros, one repeated real zero (two equal zeros), or no real zeros (two complex conjugate zeros).
• A cubic polynomial (degree 3), of the form $p(x) = ax^3 + bx^2 + cx + d$ where $a \neq 0$, has at most three zeros. It can have three distinct real zeros, one distinct real zero and a repeated real zero, one distinct real zero and two complex conjugate zeros, or three equal real zeros.
• In general, a polynomial of degree $n$ has at most $n$ real zeros. According to the Fundamental Theorem of Algebra, a polynomial of degree $n \ge 1$ has exactly $n$ complex zeros (counting multiplicity).

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Geometrical Meaning of the Zeros of a Polynomial

The relationship between algebra and geometry is profound. Any polynomial $p(x)$ can be represented geometrically by its graph in a coordinate plane. The graph of the equation $y = p(x)$ provides a visual representation of the polynomial's behavior. Most importantly, the real zeros of the polynomial are directly visible on its graph.

Real Zeros and the X-Axis

By definition, a real number 'a' is a zero of a polynomial $p(x)$ if $p(a) = 0$. When we plot the graph of $y = p(x)$, the points on the graph have coordinates $(x, y)$ where the y-value is given by the polynomial, i.e., $y = p(x)$.

If 'a' is a zero, then $p(a) = 0$, which means the point $(a, 0)$ lies on the graph. Any point with a y-coordinate of 0 lies on the x-axis.

Therefore, the conclusion is fundamental:

The real zeros of a polynomial are the x-coordinates of the points where its graph intersects or touches the x-axis.

Graph of a Linear Polynomial

A linear polynomial is of the form $p(x) = ax + b$, where $a \neq 0$. The graph of $y = ax + b$ is always a straight line.

A non-horizontal straight line will always cross the x-axis at exactly one point. The x-coordinate of this intersection is the single zero of the linear polynomial, which can be found algebraically as $x = -\frac{b}{a}$.

The graph shows a straight line crossing the x-axis at one point, indicating it has exactly one zero.

Graph of a Quadratic Polynomial

A quadratic polynomial is of the form $p(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of $y = ax^2 + bx + c$ is a U-shaped curve called a parabola.

• If $a > 0$ (positive), the parabola opens upwards ($\cup$).
• If $a < 0$ (negative), the parabola opens downwards ($\cap$).

Depending on its position, a parabola can interact with the x-axis in three possible ways, corresponding to the number of real zeros:

Case 1: Two distinct real zeros.

The graph intersects the x-axis at two different points.

Case 2: One real zero (or two equal zeros).

The graph touches the x-axis at exactly one point (the vertex of the parabola is on the axis).

Case 3: No real zeros.

The graph is entirely above or entirely below the x-axis and never intersects it. The zeros in this case are complex numbers.

Graph of a Cubic Polynomial and Higher Degrees

A cubic polynomial ($p(x) = ax^3 + ...$) has a more complex, S-shaped curve. It can intersect the x-axis at one, two, or three points. Therefore, a cubic polynomial can have 1, 2, or 3 real zeros.

In general, for a polynomial $p(x)$ of degree $n$, its graph can intersect the x-axis at at most $n$ points. This means a polynomial of degree $n$ can have at most $n$ real zeros.

Finding Information from a Graph

By observing the graph of a polynomial, we can deduce key information about it, primarily the number of real zeros and an estimate of its minimum degree.

Method to Find the Number of Zeros

This is a direct observation. The number of real zeros of the polynomial is equal to the number of times its graph intersects or touches the x-axis. Simply count these points.

Example A:

The graph intersects the x-axis at 2 distinct points. Therefore, the polynomial has 2 real zeros.

Example B:

The graph intersects the x-axis at 3 distinct points. Therefore, the polynomial has 3 real zeros.

Example C:

The graph touches the x-axis at exactly 1 point. Therefore, the polynomial has 1 real zero (which is a repeated root).

Example D:

The graph never intersects or touches the x-axis. Therefore, the polynomial has 0 real zeros.

Method to Estimate the Minimum Degree

The exact degree of a polynomial cannot always be determined from its graph, but we can find the minimum possible degree using two rules. You must check both and take the higher result.

Rule 1 (Based on Zeros): If the graph has $m$ real zeros, the degree of the polynomial, $n$, must be at least $m$.

Degree $(n) \geq$ Number of Real Zeros $(m)$

Rule 2 (Based on Turns): A "turn" on a graph is a point where it changes direction from increasing to decreasing (a peak) or vice versa (a valley). If the graph has $k$ turns, the degree, $n$, must be at least $k+1$.

Degree $(n) \geq$ Number of Turns $(k) + 1$

Applying the Degree Estimation Method to Examples

Let's apply these two rules to the previous examples to see how they work together.

Example A Analysis:

• Zeros (m): The graph has 2 real zeros.
• Turns (k): The graph has 1 turn (a single valley).

Applying the Rules:

• Rule 1: Degree $n \geq 2$ (based on 2 zeros).
• Rule 2: Degree $n \geq 1 + 1 = 2$ (based on 1 turn).

Both rules suggest the degree is at least 2. Therefore, the minimum possible degree is 2.

Example B Analysis:

• Zeros (m): The graph has 3 real zeros.
• Turns (k): The graph has 2 turns (one peak and one valley).

Applying the Rules:

• Rule 1: Degree $n \geq 3$ (based on 3 zeros).
• Rule 2: Degree $n \geq 2 + 1 = 3$ (based on 2 turns).

Both rules suggest the degree is at least 3. Therefore, the minimum possible degree is 3.

Example C Analysis:

• Zeros (m): The graph has 1 real zero.
• Turns (k): The graph has 1 turn (a single peak).

Applying the Rules:

• Rule 1: Degree $n \geq 1$ (based on 1 zero).
• Rule 2: Degree $n \geq 1 + 1 = 2$ (based on 1 turn).

Here, the rules give different minimums (1 and 2). We must choose the higher value to satisfy both conditions. Therefore, the minimum possible degree is 2. This shows why the 'turns' rule is crucial; relying only on zeros would give an incorrect estimate.

Example D Analysis:

• Zeros (m): The graph has 0 real zeros.
• Turns (k): The graph has 1 turn (a single valley).

Applying the Rules:

• Rule 1: Degree $n \geq 0$ (based on 0 zeros). This rule isn't very helpful on its own.
• Rule 2: Degree $n \geq 1 + 1 = 2$ (based on 1 turn).

Again, we must choose the higher value. The minimum degree must be at least 2. Therefore, the minimum possible degree is 2. This is another clear example of why the 'turns' rule is essential, especially when a graph has few or no real zeros.

Worked Examples

Example 1. For the polynomial represented by the graph below, find the number of zeros and the minimum possible degree.

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Example 2. For the polynomial represented by the graph below, find the number of zeros and the minimum possible degree.

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Example 3. For the polynomial represented by the graph below, find the number of zeros and the minimum possible degree.

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Example 4. For the polynomial represented by the graph below, find the number of zeros and the minimum possible degree.

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Relationship Between Zeros and Coefficients of a Polynomial

For polynomials of degree 1, 2, or 3, there are specific relationships that connect the values of their zeros to the values of their coefficients. These relationships are derived directly from the structure of the polynomial and are very useful in solving problems related to finding sums/products of zeros or constructing polynomials.

For a Quadratic Polynomial

Let the quadratic polynomial be $p(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are coefficients with $a \neq 0$. A quadratic polynomial has at most two real zeros. Let's denote these zeros as $\alpha$ (alpha) and $\beta$ (beta).

If $\alpha$ and $\beta$ are the zeros of $p(x)$, this means that when $x = \alpha$ or $x = \beta$, the value of the polynomial is zero ($p(\alpha) = 0$ and $p(\beta) = 0$). From the Factor Theorem, if $x=a$ is a zero of $p(x)$, then $(x-a)$ is a factor of $p(x)$. Therefore, if $\alpha$ and $\beta$ are the zeros, then $(x - \alpha)$ and $(x - \beta)$ are factors of $p(x)$.

So, $p(x)$ can be written as a product of these factors and a constant $k$:

Expand the right side of the equation:

So, we have $ax^2 + bx + c = k(x^2 - (\alpha + \beta)x + \alpha\beta)$. Since this equality holds for all values of $x$, the coefficients of corresponding powers of $x$ on both sides must be proportional. By comparing the leading coefficients, we can see that $k$ must be equal to $a$.

Distribute $a$ on the right side:

Now, compare the coefficients of the terms with the same power of $x$ on both sides of the equation:

• Coefficient of $x^2$: $a = a$ (This is consistent)
• Coefficient of $x$: $b = -a(\alpha + \beta)$
• Constant term: $c = a(\alpha\beta)$

From the coefficient of $x$, we can find the sum of the zeros:

From the constant term, we can find the product of the zeros:

These two formulas establish the relationship between the zeros and coefficients of a quadratic polynomial.

Converse: If the sum of the zeros ($\alpha + \beta = S$) and the product of the zeros ($\alpha\beta = P$) of a quadratic polynomial are known, then the polynomial can be written in the form $k(x^2 - Sx + P)$ for any non-zero constant $k$. If the leading coefficient is specified as $a$, the polynomial is $a(x^2 - Sx + P)$.

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For a Cubic Polynomial

Let the cubic polynomial be $p(x) = ax^3 + bx^2 + cx + d$, where $a, b, c$, and $d$ are coefficients with $a \neq 0$. A cubic polynomial has at most three real zeros. Let's denote these zeros as $\alpha$ (alpha), $\beta$ (beta), and $\gamma$ (gamma).

If $\alpha, \beta$, and $\gamma$ are the zeros of $p(x)$, then $(x - \alpha)$, $(x - \beta)$, and $(x - \gamma)$ are factors of $p(x)$.

So, $p(x)$ can be written in the form:

where $k$ is a constant (which must be equal to $a$). Expand the right side:

Now multiply $(x - \alpha)$ by the quadratic factor:

Group terms by powers of $x$:

So, we have

$$ax^3 + bx^2 + cx + d = a x^3 - a(\alpha + \beta + \gamma)x^2 + a(\alpha\beta + \beta\gamma + \gamma\alpha)x - a(\alpha\beta\gamma)$$

Comparing the coefficients of corresponding powers of $x$ on both sides:

• Coefficient of $x^3$: $a = a$ (Consistent)
• Coefficient of $x^2$: $b = -a(\alpha + \beta + \gamma)$
• Coefficient of $x$: $c = a(\alpha\beta + \beta\gamma + \gamma\alpha)$
• Constant term: $d = -a(\alpha\beta\gamma)$

From these comparisons, we get the relationships between the zeros and coefficients of a cubic polynomial:

• Sum of the zeros:

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

... (1)

• Sum of the products of zeros taken two at a time:

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

... (2)

• Product of the zeros:

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

... (3)

Converse: If the sum of zeros ($S_1 = \alpha + \beta + \gamma$), the sum of products of zeros taken two at a time ($S_2 = \alpha\beta + \beta\gamma + \gamma\alpha$), and the product of zeros ($S_3 = \alpha\beta\gamma$) are known, then a cubic polynomial can be written in the form $k(x^3 - S_1 x^2 + S_2 x - S_3)$ for any non-zero constant $k$. If the leading coefficient is specified as $a$, the polynomial is $a(x^3 - S_1 x^2 + S_2 x - S_3)$.

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Division Algorithm for Polynomials

Just as you can divide one number by another to get a quotient and a remainder (for instance, $25 \div 4$ gives a quotient of 6 and a remainder of 1, so $25 = 4 \times 6 + 1$), a similar process exists for polynomials. The Division Algorithm for Polynomials is a fundamental theorem that formalizes this procedure.

Statement of the Division Algorithm

The Division Algorithm for Polynomials states that for any two polynomials, a dividend $p(x)$ and a non-zero divisor $g(x)$, there exist unique polynomials $q(x)$ (the quotient) and $r(x)$ (the remainder) such that:

The crucial condition is that the remainder, $r(x)$, must either be the zero polynomial ($r(x) = 0$) or its degree must be strictly less than the degree of the divisor, $g(x)$.

Condition: $r(x) = 0$ or degree($r(x)$) < degree($g(x)$)

This condition is what tells us when to stop the long division process.

• $p(x)$ is the Dividend (the polynomial being divided).
• $g(x)$ is the Divisor (the polynomial we are dividing by).
• $q(x)$ is the Quotient (the main result of the division).
• $r(x)$ is the Remainder (what is left over).

The Method of Polynomial Long Division

To find the quotient $q(x)$ and remainder $r(x)$, we use a method similar to long division for numbers. Here are the steps:

1. Arrange: Write both the dividend and the divisor in standard form (descending powers of the variable). If any term is missing, insert it with a coefficient of 0. For example, write $x^3 - 2$ as $x^3 + 0x^2 + 0x - 2$.
2. Divide: Divide the first term of the dividend by the first term of the divisor. This gives the first term of the quotient.
3. Multiply: Multiply the entire divisor by this first term of the quotient.
4. Subtract: Write this product below the dividend and subtract it to get a new polynomial. (Be very careful with signs!).
5. Repeat: Treat the new polynomial as the new dividend and repeat steps 2-4. Continue this process until the degree of the remainder is less than the degree of the divisor.

Worked Examples

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Application: Finding Zeros of a Polynomial

The division algorithm is a powerful tool for finding the zeros of a polynomial when one or more zeros are already known.

• If you know one zero $x=a$, then $(x-a)$ is a factor. You can divide the original polynomial by $(x-a)$ to get a quotient of a smaller degree, which is easier to solve.
• If you know two zeros $x=a$ and $x=b$, then $(x-a)(x-b)$ is a factor. You can divide by this quadratic factor to find the remaining zeros.

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