Topics
9. Polynomials and Factoring
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United States of America
Grades 9-12

9.01 Adding and subtracting polynomials

Lesson
Practice

Introduction

We will build on our knowledge of the properties of operations with integers and apply it to a new context: polynomials. In this lesson, we will recognize and combine like terms when adding and subtracting polynomials.

Adding and subtracting polynomials

Classifying polynomials is important before performing operations on them. The following definitions will be useful in discussing the various polynomials utilized in this and further lessons:

Polynomial

The sum or difference of terms which have variables raised to non-negative integer powers and which have coefficients that are constant

Exploration

Compare\begin{aligned} & & 2x^3 & &+&4x^2 & + & 0x & +5 \\ + & & 0x^3& &+ & 3x^2& + & 2x & +3 \\ \hline \\ & & 2x^3 & & + & 7x^2 & + & 2x & +8 \end{aligned}to\begin{aligned} & & 2 && 4 && 0 && 5 \\ + & & && 3 && 2 && 3 \\ \hline \\ & & 2 && 7 && 2 && 8 \end{aligned}

  1. Create an addition problem like the example provided where the sum of the coefficients is greater than 9. What happens?

  2. Create and solve a subtraction problem using the vertical algorithm. Do polynomials behave the same as numbers when subtracting?

If the sum of the coefficients in a polynomial is greater than 9, the total will remain with the term instead of carrying over, as we see with the addition of integers. Since this is the case with adding polynomials, we may subtract polynomials similarly, and the difference may be written as a negative term instead of borrowing.

Polynomial addition leads to combining like terms, and polynomial subtraction is equivalent to adding the negative terms.

Standard form (of a polynomial)

A way of writing a polynomial expression; a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_1 x + a_0, where n is a non-negative integer and each a_i is a coefficient.

Degree (of a polynomial)

The value of the highest exponent on a variable in the polynomial

Leading term

The term in a polynomial with the highest exponent of the variable

Leading coefficient

The coefficient of the leading term

Monomial

A polynomial with one term

Binomial

A polynomial with two terms

Trinomial

A polynomial with three terms

Examples

Example 1

Consider the polynomial

3x - 6 + x^{2}

a

Rewrite the expression in standard form.

Worked Solution
Create a strategy

Recall that the standard form of a polynomial is written with the terms in order from the term with the highest variable exponent to the lowest. We can use the commutative property to change the order of the polynomials.

Apply the idea

x^{2} + 3x - 6

b

State the degree of the polynomial.

Worked Solution
Apply the idea

x^{2} + 3x - 6 is a polynomial of degree 2.

Reflect and check

Since the polynomial has 3 terms, it may be called a trinomial.

c

Identify the quadratic term, the linear term, and the constant term of the polynomial.

Worked Solution
Apply the idea
  • Quadratic term: x^{2}
  • Linear term: 3x
  • Constant term: -6
Reflect and check

Polynomials of degree 2 are called quadratic polynomials.

Example 2

Consider the polynomials x^3-6x+2 and x^2+9x+7.

a

Find the sum of the two polynomials.

Worked Solution
Create a strategy

Since we want to find the sum of the two polynomials, combine the like terms.

Apply the idea
\displaystyle \text{Sum}\displaystyle =\displaystyle \left(x^3-6x+2\right)+\left(x^2+9x+7\right)Add the polynomials together
\displaystyle =\displaystyle x^3-6x+2+x^2+9x+7Remove the parentheses (Associative Property)
\displaystyle =\displaystyle x^3+x^2+3x+9Combine the like terms
Reflect and check

If we want to use the vertical algorithm method, we need to make sure we correctly align the like terms.

\begin{aligned} & & x^3 & & & & - & 6x & + 2 \\ + & & & & & x^2 & + & 9x & + 7 \\ \hline \\ & & x^3 & & + & x^2 & + & 3x & + 9 \end{aligned}

b

Explain why the sum of two polynomials is also a polynomial.

Worked Solution
Apply the idea

By definition, a polynomial is the sum or difference of terms which have variables raised to non-negative integer powers and which have coefficients that may be real or complex.

Adding one polynomial to the other is the same as adding more terms to one polynomial. This doesn't change the fact that it is a polynomial, so the sum of two polynomials will always be a polynomial.

Reflect and check

We can use the same explanation for why the difference between two polynomials is also a polynomial, and we can extend this explanation to include the sum or difference of any number of polynomials.

Example 3

Simplify the expression:\left(3x^2-5x+1\right)-\left(x^2+7x-10\right)

Worked Solution
Apply the idea
\displaystyle \left(3x^2-5x+1\right)-\left(x^2+7x-10\right)\displaystyle =\displaystyle 3x^2-5x+1-x^2-7x+10Distribute the subtraction
\displaystyle =\displaystyle \left(3x^2-x^2\right)+\left(-5x-7x\right)+\left(1+10\right)Group the like terms together
\displaystyle =\displaystyle 2x^2-12x+11Simplify
Reflect and check

A color-coded visualization helps confirm that our distribution is correct.

A color coded visualization showing how to simplify the expression left parenthesis 3 x squared minus 5 x plus 1 right parenthesis minus left parenthesis x squared plus 7 x minus 10 right parenthesis. Speak to your teacher for more information.

Example 4

Write at least two equivalent expressions for the length of a fence around a rectangular yard with a length of 2x^{3} + 5 feet and a width of x^{2} + 6 feet.

Worked Solution
Create a strategy

Draw a diagram of the yard and label its length and width. Use the length and width to write the perimeter.

A rectangle with a width of x squared plus 6 feet and a length of 2 x cubed plus 5 feet.
Apply the idea

The perimeter of the yard is the sum of the sides. One expression for the length of the fence would be

(2x^{3} + 5) + (x^{2} + 6) +(2x^{3} + 5) + (x^{2} + 6) \text{ ft}

By finding the sum of the binomials representing the length and width of the yard, another expression would be

2x^{2}+22+4x^{3} \text{ ft}

The expression written in standard form begins with the term with the highest degree and ends with the term with the lowest degree, shown below:

4x^{3} + 2x^{2} + 22 \text{ ft}

Idea summary

We add polynomials by combining like terms. We subtract polynomials by adding the negative terms.

Outcomes

A.SSE.A.1

Interpret expressions that represent a quantity in terms of its context.

A.SSE.A.1.A

Interpret parts of an expression, such as terms, factors, and coefficients.

A.APR.A.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.

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