# Polynomial

Here you will learn about polynomials, including binomials, quadratics, cubics, and quartics. You will learn how to write them in standard form and simplify them.

Students first learn about polynomials in algebra and expand their knowledge as they progress through high school mathematics.

## What is a polynomial?

A polynomial is an algebraic expression that is made up of variables, constants, and exponents that are joined together using mathematical operations (addition, subtraction, multiplication, and division). The exponents of polynomial expressions must be whole numbers and the coefficients must be real numbers.

Polynomials are typically written in standard form which is the descending order of exponents. The leading term always has the highest exponent and defines the degree of the polynomial. The degree corresponds to the number of roots a polynomial has.

There are several classifications of polynomials by the number of terms they have.

For example,

• Monomial – this is a single polynomial expression such as 3 x^2 or \cfrac{3}{4} \, y
• Binomial – this is a two term polynomial expression such as (2 x+6) or (x-y)
• Trinomial – this is a three term polynomial expression such as \left(3 x^2-6 x+1\right)

An example of a polynomial is a quadratic with three terms, x^2+3 x-6.

The prefix “poly” means many, so polynomial expressions have many terms.

This polynomial has a variable x.

The degree of a polynomial refers to the highest power (exponent) of its variable.

• Linear polynomials have a degree of 1.
For example, 3 x+2.

• Quadratic polynomials have a degree of 2.
For example, x^2+2 x-3.

• Cubic polynomials have a degree of 3.
For example, 2 x^3-9 x^2+5-1.

• Quartic polynomials have a degree of 4.
For example, 5 x^2+x^4-3 x^3+x.

Putting polynomials in standard form helps to identify the degree and the leading coefficient.

The standard form of a polynomial is when the exponents of the variables are in descending order. The exponent of the leading term (or first term) of a polynomial in standard is the degree and the coefficient of the leading term.

For example, 5 x^2-8 x^4+7 x-10 is not in standard form. Notice how the exponents of the variable, x, are not in descending order. You can rewrite the polynomial to place it in standard form.

5 x^2-8 x^4+7 x-10 \rightarrow-8 x^4+5 x^2+7 x-10 (Standard Form)

The leading term of the polynomial is 5 ^x4. The degree of this polynomial is 4 which is the exponent of the first term. It also happens to be the highest exponent or highest power of the polynomial. The leading coefficient is -8 which is the coefficient of the first term in standard form.

Sometimes you need to expand polynomials in order to determine their degree. You can use a multiplication grid OR the distributive property to expand the polynomial and place it in standard form.

Let’s look at an example that demonstrates both methods.

What is the degree and leading coefficient of the polynomial formed from (x-6)^2\left(3 x^2+1\right) ?

You can rewrite this as (x-6)(x-6)\left(3 x^2+1\right).

Let’s first use the distributive property to multiply the first two sets of binomials.

\begin{aligned}& (x-6)(x-6) \\\\ & =x^2-6 x-6 x+36 \\\\ & =x^2-12 x+36\end{aligned}

Now, place this quadratic trinomial in brackets and multiply it by the last binomial.

\left(x^2-12 x+36\right)\left(3 x^2+1\right)

Continue using the distributive property, first multiply x^2 to 3 x^2 and +1.

3 x^4+x^2

Then take the -12x and multiply it to 3 x^2 and +1.

-36 x^3-12 x

Then take the 36 and multiply it to 3 x^2 and +1.

108 x^2+36

List all the multiplied terms out,

3 x^4+x^2-36 x^3-12 x+108 x^2+36

Combine like terms and write the polynomial in standard form.

\begin{aligned}&3 x^4+x^2-36 x^3-12 x+108 x^2+36 \\\\ &3 x^4-36 x^3+109 x^2-12 x+36\end{aligned}

The first term in the polynomial is 3 x^4 which has an exponent of 4. So, this is a polynomial of degree 4. It is a quartic.

Here, the coefficient of the x^4 term is the positive integer 3.

The coefficient of the x^3 term is -36.

The coefficient of the x^2 term is 109.

The coefficient of the x term is -12.

The constant term is 36.

OR

From this point, \left(x^2-12 x+36\right)\left(3 x^2+1\right) , you can use a multiplication grid to help keep track of terms.

Whichever method you use, be sure to write the final polynomial in standard form and combine like terms.

In this case, the polynomial in standard form is:

3 x^4-36 x^3+109 x^2-12 x+36

The degree of the polynomial is 4.

The polynomial has 5 terms.

Note: Polynomials can have coefficients that are integers, decimals, fractions or radicals.

For example, \cfrac{1}{4} x^3-\sqrt{2} x^2+0.21 x-7.

## Common Core State Standards

How does this relate to 7 th grade math and high school math?

• Grade 7 – Expressions and Equations (7.EE.1)
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

• High School Algebra : Arithmetic with Polynomials and Rational Expressions (A-APR-1)
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

## How to determine the degree and leading coefficient of a polynomial

In order to determine the degree and leading coefficient of a polynomial.

1. Rewrite the polynomial in descending order of exponents.
2. Identify the exponent of the first term and the coefficient.
3. Label the degree and the leading coefficient.

## Polynomial examples

### Example 1: write in standard form

Identify the degree and the leading coefficient of the polynomial.

5 x-x^4+9 x^5-7 x^2-11

1. Rewrite the polynomial in descending order of exponents.

9 x^5-x^4-7 x^2+5 x-11

2Identify the exponent of the first term and the coefficient.

The first term of the polynomial is 9 x^5 .

The exponent of the first term is 5 and the coefficient is 9.

3Label the degree and the leading coefficient.

The degree of the polynomial, 9 x^5-x^4-7 x^2+5 x-11, is 5 and the leading coefficient is 9 .

### Example 2: write in standard form to find degree and leading coefficient

Identify the degree and the leading coefficient of the polynomial.

-x^3-x^6+x^2-7 x-11+x^5

Rewrite the polynomial in descending order of exponents.

Identify the exponent of the first term and the coefficient.

Label the degree and the leading coefficient.

## How to write a polynomial in standard form

In order to write a polynomial in standard form:

1. Write out the terms in individual pairs of parentheses.
2. Expand the pairs of parentheses.
3. Write out the terms and combine like terms.
4. Repeat the process if there is more than one set of parentheses.
5. Write the polynomial in standard form.

### Example 3: expanding two sets of brackets

Expand the polynomial to determine the degree.

(2 x-3)(x+5)

Write out the terms in individual pairs of parentheses.

Expand the pairs of parentheses.

Write out the terms and combine like terms.

Repeat the process if there is more than one set of parentheses.

Write the polynomial in standard form.

### Example 4: squaring a pair of brackets

Expand the polynomial to determine the degree.

(3 x+4)^2

Write out the terms in individual pairs of parentheses.

Expand the pairs of parentheses – Repeat if there is more than one set.

Write out the terms and combine like terms.

Repeat the process if there is more than one set of parentheses.

Write the polynomial in standard form.

### Example 5: expanding triple brackets

Expand the polynomial to determine the degree.

(2 x+5)(x-2)(x+1)

Write out the terms in individual pairs of parentheses.

Expand the pairs of parentheses.

Write out the terms and combine like terms.

Repeat the process if there is more than one set of parentheses.

Write the polynomial in standard form.

### Example 6: cubing a pair of brackets

Expand the polynomial to determine the degree.

(2 x-3)^3

Write out the terms in individual pairs of parentheses.

Expand the pairs of parentheses.

Write out the terms and combine like terms.

Repeat the process if there is more than one set of parentheses.

Write the polynomial in standard form.

### Teaching tips for polynomial

• Use visual tools such as algebra tiles or digital algebra tiles so students can develop a concrete understanding.

• Incorporate game playing or scavenger hunts as a way for students to practice instead of just giving a worksheet.

### Easy mistakes to make

• Incorrect signs when multiplying or combining terms with negative numbers
A common error is to forget negative signs when expanding double parentheses.
Using a grid method and writing the signs next to each term can help with this.
For example,

• Forgetting the middle terms when expanding double parentheses
A common error is to forget to find all the terms when expanding double brackets.
Using a grid method can help you get the correct amount of terms.

### Practice polynomial questions

1. Identify the degree and the leading coefficient of the polynomial,

-4 x^3+6 x^5-10-x^4+x

Degree: 3 and Leading coefficient; -4

Degree: -4 and Leading coefficient: 3

Degree: 5 and Leading coefficient: 6

Degree: 6 and Leading coefficient: 5

First rewrite the polynomial in standard form which is in descending order of exponents.

-4 x^3+6 x^5-10-x^4+x \rightarrow 6 x^5-x^4-4 x^3+x-10

The first term in standard form is 6 x^5. The degree is the exponent of the first term which in this case is 5 and the leading coefficient is the coefficient of the first term which is 6.

2. Identify the degree and the leading coefficient of the polynomial,

9+8 x^2-x^3-\cfrac{1}{2} \, x

Degree: 0 and Leading coefficient; 9

Degree: 3 and Leading coefficient: 1

Degree: 1 and Leading coefficient: 3

Degree: 3 and Leading coefficient: -1

First rewrite the polynomial in standard form which is in descending order of exponents.

9+8 x^2-x^3-\cfrac{1}{2} \, x \rightarrow-x^3+8 x^2-\cfrac{1}{2} \, x+9

The first term in standard form is -x^3. The degree is the exponent of the first term which in this case is 3 and the leading coefficient is the coefficient of the first term which is -1.

3. Expand the polynomial and simplify.

(2 x-1)(2 x+5)

4x+4

4 x^2-5

4 x^2+8x-5

4 x^2+11x-5

Using a multiplication grid:

OR

You can use the distributive property:

First multiply 2x to 2x and +5 \rightarrow 4 x^2+10 x

Then multiply -1 to 2 and +5 \rightarrow -2 x-5

4 x^2+10 x-2 x-5,10 x and -2 x are like terms. They combine to be 8x.

10 x-2 x=8 x

The polynomial in standard form is: 4 x^2+8 x-5.

4. Expand the polynomial and simplify.

(4 x+3)^2

16 x^2+24 x+9

16 x^2+9

8x+6

16 x^2+12 x+9

Using the multiplication grid:

(4 x+3)^2=(4 x+3)(4 x+3)

Using the distributive property:

(4 x+3)^2=(4 x+3)(4 x+3)

First take 4x and multiply it to 4x and +3 \rightarrow 16 x^2+12 x

Then take 3 and multiply it to 4x and +3 \rightarrow 12x+9

16 x^2+12 x+12 x+9,12 x and 12 x are like terms that combine to be 24x.

12 x+12 x=24 x

The polynomial in standard form is: 16 x^2+24 x+9.

5. Expand the polynomial and simplify.

(x-4)(2 x+1)(3 x-2)

6x^3-25x^2+2x+8

6 x^3+8

6x-5

6x^3-25x^2+26x+8

Using the multiplication grid to expand:

You can also use the distributive property, but in this case the multiplication grid organizes all the terms better.

6. Expand the polynomial and simplify.

(x+3)^2(2 x-1)

2 x^3+12 x^2+11 x-9

2 x^3+11 x^2+12 x-9

2 x^3+13 x^2+13 x+9

2 x^3+11x^2+12 x+9

Using the multiplication grid to expand:

(x+3)^2=(x+3)(x+3)

You can use the distributive property to expand too.

## Polynomial FAQs

Can you solve polynomial equations?

Yes, you can solve polynomial equations using several strategies such as factoring, graphing, synthetic division, and/or using long division.

What are rational functions?

A rational function is a ratio (or quotient) of polynomials where the polynomial in the denominator cannot be equal to 0, for example divide polynomials p and q (\frac{polynomial \text { p }} {polynomial \text { q }}), where polynomial q 0.

What is the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra states that polynomial equations of degree n where n \geq 1 with complex number coefficients has at least one root in the complex number system.

Can a polynomial function have a negative exponent?

No, polynomials have only non-negative integer exponents and 0.

What is a zero polynomial?

A zero polynomial is a polynomial that has 0 ’s as its coefficients.

Is a quadratic equation considered a polynomial equation?

Yes, a quadratic equation is a polynomial equation with the highest degree of 2.

Are linear equations considered to be polynomial equations?

Yes, linear equations are polynomials equations with degree 1.

Are exponential functions considered polynomial functions?

No, exponential functions are not polynomials because the exponent is not always a whole number.

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