Polynomials

A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Below are some examples of polynomials:

$$\begin{array}{c}&x+3, &3x^2-2x+5, &-7, &2a^3b^2-3b^2+2a-1, &\frac{1}{2}x^2-\frac{2}{3}x+\frac{3}{4}. \end{array}$$

Polynomials are an important part of the "language" of mathematics and algebra. They are used in nearly every field of mathematics to express numbers as a result of mathematical operations. Polynomials are also "building blocks" in other types of mathematical expressions, such as rational expressions.

Many mathematical processes that are done in everyday life can be interpreted as polynomials. Summing the cost of items on a grocery bill can be interpreted as a polynomial. Calculating the distance traveled of a vehicle or object can be interpreted as a polynomial. Calculating perimeter, area, and volume of geometric figures can be interpreted as polynomials. These are just some of the many applications of polynomials.

A polynomial is a special type of mathematical expression.

A mathematical expression is a number represented by variables, constants, and the mathematical operations performed on them.

Below are some examples of expressions:

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$$\begin{array}{ccccc} \color{#3D99F6}{3x^2-2x+5} & \hphantom{\ldots} & \color{#3D99F6}{\frac{1}{2}x^2-\frac{2}{3}x+\frac{3}{4}} & \hphantom{\ldots} & \color{#D61F06}{2^x+x^{1/2}} \\ \\ \color{#D61F06}{\frac{x}{y}+2y} & \hphantom{\ldots} & \color{#D61F06}{6x^{-2}+2x-3} & \hphantom{\ldots} & \color{#3D99F6}{x+3} \\ \\ \color{#D61F06}{\cos(x^2-1)} & \hphantom{\ldots} & \color{#3D99F6}{2a^3b^2-3b^2+2a-1} & \hphantom{\ldots} & \color{#3D99F6}{-7} \\ \\ \end{array}$$

Some of the expressions above are polynomials (in blue), and some are not (in red). The polynomials can be identified by noting which expressions contain only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The non-polynomial expressions will be the expressions which contain other operations.

Explain why the non-polynomial expressions are not polynomials.

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We can summarize the reasons as in the following table:

$$\begin{array}{|c|c|} \hline \text{Non-Polynomial Expression} & \text{Reason it is not a polynomial} \\ \hline 2^{\color{#D61F06}{x}}+x^{\color{#D61F06}{1/2}} & \text{Polynomials cannot contain variable exponents.} \\ & \text{They also cannot contain non-integer exponents.} \\ \hline \frac{x}{\color{#D61F06}{y}}+2y & \text{In general, polynomials }can\text{ contain fractions.} \\ & \text{However, they cannot contain variables in a denominator.} \\ \hline 6x^{\color{#D61F06}{-2}}+2x-3 & \text{Polynomials cannot have negative exponents on variables.} \\ \hline \color{#D61F06}{\cos}(x^2-1) & \text{Polynomials cannot contain non-polynomial functions}\\ &\text{including trigonometric functions like cosine.} \\ \hline \end{array}$$

Polynomials are well-understood mathematical objects, so it is convenient for mathematicians to be able to express mathematical processes as polynomials. Non-polynomial expressions tend to present more challenges when solving mathematical problems. There is a concept in calculus, called a Taylor series approximation, in which the goal is to approximate a non-polynomial expression as a polynomial expression. This is done because of the many convenient properties of polynomials.

The vocabulary involved in polynomials can be a bit intimidating at first. However, these "complicated-sounding" words are often used to represent simple ideas.

The "building blocks" of polynomials are called monomials.

A monomial is a polynomial expression that contains variables and a coefficient, and does not contain addition or subtraction.

Monomials are often called terms if they are a part of a larger polynomial.

Identify the terms in each polynomial.

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We can summarize the answers as follows:

$$\begin{array}{|c|c|} \hline \text{Polynomial Expression} & \text{Terms} \\ \hline x+3 & x\text{ and }3 \\ \hline 3x^2-2x+5 & 3x^2\text{, }-2x\text{, and } 5 \\ \hline -7 & -7 \\ \hline 2a^3b^2-3b^2+2a-1 & 2a^3b^2\text{, }-3b^2\text{, }2a\text{, and }-1 \\ \hline \frac{1}{2}x^2-\frac{2}{3}x+\frac{3}{4} & \frac{1}{2}x^2\text{, }-\frac{2}{3}x\text{, and }\frac{3}{4} \\ \hline \end{array}$$

Note that each term can be positive or negative, and this sign depends on whether the term was added in the polynomial or subtracted in the polynomial. Each term also has a coefficient.

The coefficient of a term is the non-variable factor of that term.

Identify the coefficient in each term.

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The answers are as follows:

$$\begin{array}{|c|c|} \hline \text{Term} & \text{Coefficient} \\ \hline x & 1 \\ \hline 3x^2 & 3 \\ \hline -2x & -2 \\ \hline 2a^3b^2 & 2 \\ \hline -\frac{2}{3}x & -\frac{2}{3} \\ \hline -7 & -7 \\ \hline \end{array}$$

Note that the "default" value of a coefficient is $1$. If a term contains no variables, then the coefficient is the term itself.

Polynomials are often classified by degree.

The degree of a monomial is the sum of the exponents of each variable in the monomial.

The degree of a polynomial is the largest degree out of all the degrees of monomials in the polynomial.

Identify the degree of each polynomial discussed above.

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The answers are as follows:

• $-7$: Constant monomials always have a degree of $\color{#3D99F6}0$. They could also be expressed as, for instance, $-7x^0$ as $x^0 = 1$ for any $x \neq 0$.
• $x+3$: Note that $x=x^1$. The degree of $x^{\color{#D61F06}{1}}$ is $\color{#D61F06}1$. The degree of $3$ is $\color{#D61F06}0$. The degree of the polynomial is the larger of the degrees, which is $\color{#3D99F6}1$.
• $3x^2-2x+5$: Note that $-2x=-2x^1$. The degree of $3x^{\color{#D61F06}{2}}$ is $\color{#D61F06}2$. The degree of $-2x^{\color{#D61F06}{1}}$ is $\color{#D61F06}1$. The degree of $5$ is $\color{#D61F06}0$. The degree of the polynomial is the largest of these degrees, which is $\color{#3D99F6}2$.
• $2a^3b^2-3b^2+2a-1$: Note that $2a=2a^1$. The degree of $2a^{\color{#D61F06}{3}}b^{\color{#D61F06}{2}}$ is $\color{#D61F06}3+\color{#D61F06}2=\color{#D61F06}5$. The degree of $-3b^{\color{#D61F06}{2}}$ is $\color{#D61F06}2$. The degree of $2a^{\color{#D61F06}{1}}$ is $\color{#D61F06}1$. The degree of $-1$ is $\color{#D61F06}0$. The degree of the polynomial is the largest of these degrees, which is $\color{#3D99F6}5$.
• $\frac{1}{2}x^2-\frac{2}{3}x+\frac{3}{4}$: Note that $-\frac{2}{3}x=-\frac{2}{3}x^1$. The degree of $\frac{1}{2}x^{\color{#D61F06}{2}}$ is $\color{#D61F06}2$. The degree of $-\frac{2}{3}x^{\color{#D61F06}{1}}$ is $\color{#D61F06}1$. The degree of $\frac{3}{4}$ is $\color{#D61F06}0$. The degree of the polynomial is the largest of these degrees, which is $\color{#3D99F6}2$. $_\square$

Polynomials are classified in this way because they exhibit different mathematical behavior and properties depending on what the degree is. The degree of a polynomial also affects the problem-solving strategy for solving equations containing that polynomial.

$0$ degree polynomials are called constants. The values of constants don't change, so they're used to describe quantities that don't change.

$1^\text{st}$ degree polynomials are called linear polynomials. They are used to describe quantities that change at a steady rate. They are also used in many one-dimensional geometry problems involving length.

$2^\text{nd}$ degree polynomials are called quadratic polynomials. They are used to describe quantities that change with some amount of acceleration or deceleration. They are also used in many two-dimensional geometry problems involving area.

$3^\text{rd}$ degree polynomials are called cubic polynomials. They are used in many three-dimensional geometry problems involving volume.

There are no special names for polynomials that are $4^\text{th}$ degree or higher. Higher-degree polynomials have varied applications.

The terms "constant," "linear," "quadratic," and "cubic" are common in mathematics; they are not just used in polynomials. However, the meaning of each of these words is always connected to the degree of some polynomial.

Polynomials represent numbers, and as such, any mathematical operation can be performed on polynomials just as they are done on numbers. When polynomials are added, subtracted, or multiplied, the result is another polynomial. When polynomials are divided, the result is a rational expression.

Addition and Subtraction

Main Article: Combining Like Terms

There are two polynomials: $(3x^2 -2x+4)$ and $(-3x^2+6x-10)$ What is the sum of these polynomials?

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The sum is $(3x^2 -2x+4)+(-3x^2+6x-10)$.

By the associative property of addition, the grouping of terms does not matter. Therefore, the parentheses can be eliminated without changing the result. The sum is then written as $3x^2 -2x+4+-3x^2+6x-10$.

Combining like terms, the resulting sum is $4x-6$. $_\square$

In a similar fashion, polynomials can also be subtracted.

There are two polynomials: $(2x^3+x^2+x+1)$ and $(2x^2+3x+4)$. What is the difference of these polynomials?

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The difference is $(2x^3+x^2+x+1)-(2x^2+3x+4)$.

The subtraction can be re-interpreted as a sum with the negation of the second polynomial. Re-interpreted as a sum, the expression becomes $(2x^3+x^2+x+1)+(-2x^2-3x-4)$. Now that the expression is a sum, the associative property of addition makes the grouping of terms irrelevant. The parentheses can be eliminated: $2x^3+x^2+x+1-2x^2-3x-4.$

Combining like terms, the resulting difference is $2x^3-x^2-2x-3$. $_\square$

Multiplication

Main Article: Multiplying polynomials

Multiplication of two polynomials involves multiplying each term of the first polynomial with each term of the second polynomial, and then summing the resulting monomials. When multiplying terms, one must remember the rule of product for exponents.

There are two polynomials: $(x^3+1)$ and $(x^2+1)$. What is the product of these polynomials?

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The product is written as $(x^3+1)(x^2+1).$

To show that each term in the first polynomial is multiplied by each term in the second polynomial, re-write it as $$x^3(x^2+1)+1(x^2+1).$$ Now multiplying and combining like terms, $$x^5+x^3+x^2+1.$$ There are no like terms, so the resulting product is $x^5+x^3+x^2+1$. $_\square$

Division

Main Article: Polynomial Division

Dividing polynomials often involves re-writing the division as a rational expression.

There are two polynomials: $(2x^2-3x+8)$ and $(x-3)$. Write the quotient of these polynomials as a rational expression.

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The quotient written as a rational expression is $\frac{2x^2-3x+8}{x-3}.$ $_\square$

This is often the preferred way of writing the quotient. Sometimes the resulting rational expression can be further simplified, but not in this case.

Another option to write a quotient of polynomials is to write them as the sum of a polynomial and a rational expression using polynomial division.

There are two polynomials: $(2x^2-3x+8)$ and $(x-3)$. Use polynomial division to write the quotient of these polynomials as the sum of a polynomial and a rational expression.

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The long division is as follows:

Thus, the resulting quotient is $2x+3+\frac{17}{x-3}.$ $_\square$

Main Article: Factoring polynomials

Factoring polynomials is the process of re-writing a polynomial as the equivalent product of polynomials. There are three common ways in which a polynomial can be factored: grouping, substitution, and using identities.

Factoring by Grouping:

Factor $x^3+x^2+x+1$ by grouping.

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We have

$$\begin{aligned} x^3+x^2+x+1 &= (x^3+x^2)+(x+1) \\ &= x^2(x+1)+1(x+1) \\ &= (x^2+1)(x+1).\ _\square \end{aligned}$$

Factoring by Substitution:

Factor $2{(y+1)}^2 + 6(y+1) + 4.$

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Let $x=(y+1),$ then the polynomial becomes $2x^2+6x+4 = (2x+4)(x+1).$
Substituting in $x=y+1$ gives $(2y+6)(y+2).\ _\square$

Factoring with Identities:

Factor $x^2-25.$

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Recall the difference of squares identity: $a^2-b^2=(a-b)(a+b).$ Then we have

$$x^2-25=(x-5)(x+5).\ _\square$$

Factor $x^{4} + x^{2} + 1$.

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Recall the identity: $a^2+2ab+b^2=(a+b)^2.$ If this identity were to be used, the $x^{2}$ term should have a coefficient of $2$. Make the coefficient $2$ by subtracting $x^2$ at the end:

$$x^{4} + x^{2} + 1= x^{4} + 2x^{2} + 1 -x^{2}.$$

Then factoring the perfect square trinomial gives

$$\big( x^{2} + 1 \big)^{2} - x^{2}.$$

This is now a difference of squares:

$$\big( x^{2} + 1 + x \big)\big(x^{2} + 1 - x \big).$$

Now re-order the terms by descending degree, and we have

$$\big( x^{2}+x+1 \big)\big(x^{2}-x+1 \big).\ _\square$$

Factor $x^{4} + y^{4}$.

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As before, this can be accomplished by adding a term and subtracting the same term. The desired identity is again the perfect square identity, so there should be $2 x^{2} y^{2}$ term in the middle. Add and subtract this term:

$$x^{4} + 2x^{2}y^{2} + y^{4} - 2x^{2}y^{2}.$$

Factor the perfect square trinomial:

$$\big( x^{2} + y^{2} \big)^{2} - 2x^{2}y^{2}.$$

Now factor the difference of squares:

$$\big( x^{2} + y^{2} + \sqrt{2}xy \big) \big( x^{2} + y^{2} - \sqrt{2}xy \big).\ _\square$$

A polynomial function is a function which is evaluated as a polynomial.

A function $p(x)$ is a polynomial function if it can be written as

$$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0.$$

In this form, $a_0, a_1, \cdots , a_{n-1}, a_n$ are non-variable coefficients, and $n$ is a non-negative integer.

Put more simply, a function is a polynomial function if it is evaluated with addition, subtraction, multiplication, and non-negative integer exponents. Polynomial functions can also be multivariable. For example, $q(x,y)=3x^2y+2xy-6x+9$ is a polynomial function.

Main Article: Remainder Factor Theorem

The remainder theorem and factor theorem are important results for polynomial functions involving the evaluation of those functions and the zeros of those functions, respectively.

Remainder Theorem

When a polynomial $p(x)$ is divided by $(x-a)$, the remainder is $p(a)$.

Let $p(x)$ be a polynomial function. When $p(x)$ is divided by $(x-a)$, the result will be the sum of a polynomial function and a rational expression: $$\dfrac{p(x)}{x-a}=q(x)+\dfrac{r}{x-a},$$ where $q(x)$ represents the resulting quotient polynomial, and $r$ represents the resulting remainder. Multiplying both sides of this equation by $(x-a)$ yields $$p(x)=(x-a)q(x)+r.$$ Plugging in $x=a$, we have $p(a)=(a-a)q(x)+r.$

Therefore, $r=p(a)$. $_\square$

Factor theorem

Let $f(x)$ be a polynomial function such that $f(c)=0$ for some constant $c.$ Then $(x-c)$ is a factor of $f(x)$. The converse of the statement is also true.

What is the remainder when $f(x)=x^3+5x^2+3$ is divided by $x+1?$

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Directly from the remainder theorem, the remainder is

$$f(-1)=-1+5+3=7.\ _\square$$

Main Article: Newton's Identities

Newton's identities, also called Newton's sums or the Newton-Girard formulae, give an efficient way to compute the power series of the roots of a polynomial equation without computing the roots themselves.

Let $\alpha_1$ and $\alpha_2$ be the roots of the polynomial equation

$$x^2+x+1=0.$$

What is the value of $\alpha_1^3+\alpha_2^3?$

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Since $a=1, b=1,$ and $c=1,$ by Newton's sums

$$\begin{aligned} P_0 &= a_1^0+a_2^0 = 2 \\ P_1 &= a_1^1+a_2^1 = \frac{-b}{a} = -1 \\ P_2 &= a_1^2+a_2^2 = \frac{-b}{a}P_1-\frac{c}{a}P_0 = -1 \\ P_3 &= a_1^3+a_2^3 = \frac{-b}{a}P_2-\frac{c}{a}P_1 = 2.\ _\square \end{aligned}$$