13

Polynomials

In this chapter, you learn about polynomials. The chapter begins with a discussion of the elementary concepts that you need to know to ensure your success when working with polynomials and ends with addition and subtraction of polynomials.

Terms and Monomials

In an algebraic expression, terms are the parts of the expression that are connected to the other parts by indicated addition (plus sign) or indicated subtraction (minus sign). If the algebraic expression has no indicated addition or subtraction, then the algebraic expression itself is a term.

Problem

Identify the terms in the given expression.

a.

b. 3x⁵

Solution

a.

Step 1.   The expression contains indicated addition and subtraction, so identify the quantities between the plus and minus signs.

The terms are −8x, , and 27.

b. 3x⁵

Step 1.   There is no indicated addition or subtraction, so the expression is a term.

The term is 3 x⁵.

In monomials, no variable divisors, negative exponents, or variables as radicands of simplified radicals are allowed.

A monomial is a special type of term that, when simplified, is a constant or a product of one or more variables raised to nonnegative integer powers, with or without an explicit coefficient.

Problem

Specify whether the term is a monomial. Explain your answer.

a.

b.

c. 0

d. 3x⁵

e.

f. 4x^-3y²

g.

Solution

a. −8x

Step 1.   Check whether −8x meets the criteria for a monomial.

-8x is a term that is a variable raised to a positive integer power of 1 (understood), with an explicit coefficient of −8, so it is a monomial.

Step 1.   Check whether meets the criteria for a monomial.

is a term, but it contains division by a variable, so it is not a monomial.

c. 0

Step 1.   Check whether 0 meets the criteria for a monomial.

0 is a constant, so it is a monomial.

d. 3x⁵

Step 1.   Check whether 3x⁵ meets the criteria for a monomial.

3x⁵ is a term that is a variable raised to a positive integer power of 5, with an explicit coefficient of 3, so it is a monomial.

e.

Step 1.   Check whether meets the criteria for a monomial.

is a constant, so it is a monomial.

f. 4x^-3y²

Step 1.   Check whether 4x^-3y² meets the criteria for a monomial.

4x^-3y² contains a negative exponent, so it is not a monomial.

g.

Step 1.   Check whether meets the criteria for a monomial.

is a term, but it contains a variable as the radicand of a simplified radical, so it is not a monomial.

The constants in monomials can be divisors, have negative exponents, or be radicands in a radical. For instance, is a monomial.

Polynomials

A polynomial consists of a single monomial or two or more monomials connected by plus or minus signs. A polynomial that has exactly one term is a monomial. A polynomial that has exactly two terms is a binomial. A polynomial that has exactly three terms is a trinomial. A polynomial that has more than three terms is just a general polynomial.

Problem

State the most specific name for the given polynomial.

a. x²-1

b. 8a³ + 64b³

c. x² + 4x - 12

d.

e. −2x⁵ + 5x⁴ - 3x³ - 7x² + x + 4

Solution

a. x²-1

Step 1.   Count the terms of the polynomial.

x²-1 has exactly two terms.

Step 2.   State the specific name.

x²-1 is a binomial.

b. 8a³ + 64b³

Step 1.   Count the terms of the polynomial.

8a³ + 64b³ has exactly two terms.

Step 2.   State the specific name.

8a³ + 64b³ is a binomial.

c. x² + 4x −12

Step 1.   Count the terms of the polynomial.

x² + 4x - 12 has exactly three terms.

Step 2.   State the specific name.

x² + 4x - 12 is a trinomial.

d.

Step 1.   Count the terms of the polynomial.

has exactly one term

Step 2.   State the specific name.

is a monomial.

e. −2x⁵ + 5x⁴ - 3x³ - 7x² + x + 4

Step 1.   Count the terms of the polynomial.

-2x⁵ + 5x⁴ - 3x³ - 7x² + x + 4 has exactly six terms.

Step 2.   State the specific name.

-2x⁵ + 5x⁴ - 3x³ - 7x² + x + 4 is a polynomial.

Like Terms

Monomials that are constants or monomials that have exactly the same variable factors (i.e., the same letters with the same corresponding exponents) are like terms. Like terms are the same except, perhaps, for their coefficients. Terms that are not like terms are unlike terms.

Problem

State whether the given monomials are like terms. Explain your answer.

a. −10x and 25x

b. 4x²y³ and −7x³y²

c. 100 and 45

d. 25 and 25x

Solution

a. −10x and 25x

Step 1.   Check whether −10x and 25x meet the criteria for like terms.

-10x and 25x are like terms because they are exactly the same except for their numerical coefficients.

b. 4x²y³ and −7x³y²

Step 1.   Check whether 4x²y³ and −7x³y² meet the criteria for like terms.

4x²y³ and −7x³y² are not like terms because the corresponding exponents on x and y are not the same.

c. 100 and 45

Step 1.   Check whether 100 and 45 meet the criteria for like terms.

100 and 45 are like terms because they are both constants.

d. 25 and 25x

Step 1.   Check whether 25 and 25x meet the criteria for like terms.

25 and 25x are not like terms because they do not contain the same variable factors.

Adding and Subtracting Monomials

Because variables are standing in for numbers, you rely on the properties of numbers to justify operations with polynomials. (See Chapter 1 for a discussion of the properties of numbers.)

Addition and Subtraction of Monomials

1. To add monomials that are like terms, add their numerical coefficients and use the sum as the coefficient of their common variable component.

2. To subtract monomials that are like terms, subtract their numerical coefficients and use the difference as the coefficient of their common variable component.

3. To add or subtract unlike terms, indicate the addition or subtraction.

Problem

Simplify.

a. −10x+ 25x

b. 4x²y³ - 7x³y²

c. 9x² + 3x² - 7x²

d. 25 + 25x

e. 5x² - 7x²

Solution

a. −10x+ 25x

Step 1.   Check for like terms.

-10x and 25x are like terms.

Step 2.   Add the numerical coefficients.

-10 + 25 = 15

Step 3.   Use the sum as the coefficient of x.

-10x + 25x = 15x

b. 4x²y³ - 7x³y²

-10x + 25x ≠ 15x². In addition and subtraction, the exponents on your variables do not change.

Step 1.   Check for like terms.

4x²y³ and 7x³y² are not like terms, so leave the problem as indicated subtraction: 4x²y³ - 7x³y².

c. 9x² + 3x² - 7x²

Step 1.   Check for like terms.

9x, 3x², and 7x² are like terms.

Step 2.   Combine the numerical coefficients.

9 + 3 - 7 = 5

Step 3.   Use the result as the coefficient of x².

9x² + 3x² - 7x² = 5x²

d. 25 + 25x

Step 1.   Check for like terms.

25 and 25x are not like terms, so leave the problem as indicated addition: 25 + 25x.

e. 5x² - 7x²

25 + 25x ≠ 50x. These are not like terms, so you cannot combine them into one single term.

Step 1.   Check for like terms.

5x² and 7x² are like terms.

Step 2.   Subtract the numerical coefficients.

5 - 7 = −2

Step 3.   Use the result as the coefficient of x².

5x² - 7x² = −2x²

Simplifying Polynomial Expressions

When you have an assortment of like terms in the same expression, systematically combine matching like terms in the expression. (For example, you might proceed from left to right.) You are simplifying the expression when you do this. To organize the process, use the properties of numbers to rearrange the expression so that matching like terms are together (later, you might choose to do this step mentally). If the expression includes unlike terms, just indicate the sums or differences of such terms. To avoid sign errors as you work, keep a - symbol with the number that follows it.

Problem

Simplify.

a. 4x³ + 5x² - 10x + 25 + 2x³ - 7x² - 5

b. 5x - 20x + 30 - 10x + 100 + 3x - 25

Solution

a. 4x³ + 5x² - 10x + 25 + 2x³ - 7x² - 5

Step 1.   Check for like terms.

The like terms are 4x³ and 2x³, 5x² and 7x², and 25 and 5.

Step 2.   Rearrange the expression so that like terms are together.

4x³ + 5x² - 10x + 25 + 2x³ - 7x² - 5

= 4x³ + 2x³ + 5x² - 7x² - 10x + 25 - 5

Step 3.   Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

= 6x³ + −2x² - 10x + 20

= 6x³ - 2x² −10x + 20

Step 4.   Review the main results.

4x³ + 5x² - 10x + 25 + 2x³ - 7x² - 5

= 4x³ + 2x³ + 5x² - 7x² - 10x + 25 - 5

= 6x³ - 2x² - 10x + 20

b. 5x - 20x + 30 - 10x + 100 + 3x - 25

When you are simplifying, rearranging so that like terms are together can be done mentally. However, actually writing out this step helps you avoid careless errors.

Remember, when rearranging, to keep a - symbol with the number that follows it.

Because +- is equivalent to -, it is customary to change + - to simply -when you are simplifying expressions.

You should write polynomial answers in descending powers of a variable.

Step 1.   Check for like terms.

The like terms are 5x, 20x, 10x, and 3x and 30, 100, and 25.

Step 2.   Rearrange the expression so that like terms are together.

5x - 20x + 30 - 10x + 100 + 3x - 25

= 5x - 20x - 10x + 3x + 30 + 100 - 25

Step 3.   Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

= −22x +105

Step 4.   Review the main results.

5x - 20x + 30 - 10x + 100 + 3x - 25

= 5x - 20x - 10x + 3x + 30 + 100 - 25

= −22x + 105

Adding Polynomials

Addition of polynomials involves adding like terms.

Addition of Polynomials

To add two or more polynomials, add like monomial terms and simply indicate addition of unlike terms.

Problem

Perform the indicated addition.

a. (9x² - 6x + 2) + (-7x² - 5x + 3)

b. (4x³ + 3x² - x + 8) + (8x³ + 2x - 10)

Solution

a. (9x² - 6x + 2) + (-7x² - 5x + 3)

Step 1.   Remove parentheses.

(9x² - 6x + 2) + ( 7x² - 5x + 3)

= 9x² - 6x + 2 - 7x² - 5x + 3

Step 2.   Rearrange the terms so that like terms are together. (You may do this step mentally.)

= 9x² - 7x² - 6x - 5x + 2 + 3

Step 3.   Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

= 2x² −11 x + 5

Step 4.   Review the main results.

(9x² - 6x + 2) + (-7x² - 5x + 3)

= 9x² - 6x + 2 - 7x² - 5x + 3

= 9x² - 7x² - 6x - 5x + 2 + 3

= 2x² - 11x + 5

b. (4x³ + 3x² - x + 8) + (8x³ + 2x - 10)

Step 1.   Remove parentheses.

(4x³ + 3x² - x + 8) + (8x³ + 2x - 10)

= 4x³ + 3x² - x + 8 + 8x³ + 2x - 10

Step 2.   Rearrange the terms so that like terms are together. (You may do this step mentally.)

= 4x³ + 8x³ + 3x² - x + 2x + 8 - 10

Step 3.   Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

= 12x³ + 3x² + x - 2

Step 4.   Review the main results.

(4x³ + 3x² - x + 8) + (8x³ + 2x - 10)

= 4x³ + 3x² - x + 8 + 8x³ + 2x - 10

= 4x³ + 8x³ + 3x² - x + 2x + 8 - 10

= 12x³ + 3x² + x - 2

Subtracting Polynomials

Subtraction of polynomials relies on your skills in adding polynomials.

Subtraction of Polynomials

To subtract two polynomials, add the opposite of the second polynomial.

You can accomplish subtraction of polynomials by enclosing both polynomials in parentheses and then placing a minus symbol between them. Of course, make sure that the minus symbol precedes the polynomial that is being subtracted.

Problem

Perform the indicated subtraction.

a. (9x² - 6x + 2) - (-7x² - 5x + 3)

b. (4x³ + 3x² - x + 8) - (8x³ + 2x - 10)

Solution

a. (9x² - 6x + 2) - (-7x² - 5x + 3)

Step 1.   Remove parentheses.

(9x² - 6x + 2) - (-7x² - 5x + 3)

= 9x² - 6x + 2 + 7x² + 5x - 3

Step 2.   Rearrange the terms so that like terms are together. (You may do this step mentally.)

= 9x² + 7x² - 6x + 5x + 2 - 3

Step 3.   Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

= 16x² - x - 1

Step 4.   Review the main results.

(9x² - 6x + 2) - (-7x² - 5x + 3)

= 9x² - 6x + 2 + 7x² + 5x - 3

= 9x² + 7x² - 6x + 5x + 2 - 3

= 16 x² - x - 1

b. (4x³ + 3x² - x + 8) - (8x³ + 2x - 10)

Be careful with the signs! Sign errors are common mistakes in simplifying. Be sure to change the sign of every term in the second polynomial.

Step 1.   Remove parentheses.

(4x³ + 3x² - x + 8) - (8x³ + 2x - 10)

= 4x³ + 3x² - x + 8 - 8x³ - 2x + 10

Step 2.   Rearrange the terms so that like terms are together. (You may do this step mentally.)

= 4x³ - 8x³ + 3x² - x - 2x + 8 + 10

Step 3.   Systematically combine matching like terms and indicate addition or subtraction of unlike terms.

= −4x³ + 3x² - 3x + 18

Step 4.   Review the main results.

(4x³ + 3x² - x + 8) - (8x³ + 2x - 10)

= 4x³ + 3x² - x + 8 - 8x³ - 2x + 10

= 4x³ - 8x³ + 3x² - x - 2x + 8 + 10

= −4x³ + 3x² - 3x + 18

Exercise 13

For 1-5, state the most specific name for the given polynomial.

1. x² - x + 1

2. 125x³ - 64y³

3. 2x² + 7x - 4

4.

5. 2x⁴ + 3x³ - 7x² - x + 8

For 6-15, simplify.

6. −15x + 17x

7. 14xy³ - 7x³y²

8. 10x² - 2x² - 20x²

9. 10 + 10x

10. 2 + 4(x + 5)

11. 12x³ - 5x² + 10x - 60 + 3x³ - 7x² - 1

12. (10x² - 5x + 3) + (6x² + 5x - 13)

13. (20x³ - 3x² - 2x + 5) + (9x³ + x² + 2x - 15)

14. (10x² - 5x + 3) - (6x² + 5x - 13)

15. (20x³ - 3x² - 2x + 5) - (9x³ + x² + 2x - 15)