Chapter Eleven — Polynomials

The Humongous Book of Algebra Problems

239

Note: Problems 11.5–11.6 refer to the polynomial 4x

2

– 7x + 6.

11.6 Identify the leading coefﬁcient and the constant of the polynomial.

According to Problem 11.5, the polynomial has degree two because x

2

represents the highest power of x. The coefﬁcient of x

2

is the leading coefﬁcient

of the polynomial: 4.

A constant is a term with no explicitly stated variable. Of the terms in

4x

2

– 7x + 6, the rightmost term has no variable, so 6 is the constant.

Note: Problems 11.7–11.8 refer to the polynomial ax

2

– bx

3

– c. Assume that a, b, and c are

nonzero integers.

11.7 Classify the polynomial according to its degree and the number of terms it

contains.

The polynomial consists of three terms: ax

2

, –bx

3

, and –c. The highest power of x

among the terms is 3, so the polynomial has degree three. Therefore,

ax

2

– bx

3

– c is a cubic trinomial.

Note: Problems 11.7–11.8 refer to the polynomial ax

2

– bx

3

– c. Assume that a, b, and c are

nonzero integers.

11.8 Identify the leading coefﬁcient and the constant of the polynomial.

According to Problem 11.7, the term containing the highest variable power is

–bx

3

. Its coefﬁcient, –b, is the leading coefﬁcient of the polynomial. The term –c

is a constant because it contains no variables.

Adding and Subtracting Polynomials

Only works for like terms

11.9 Simplify the expression: (5x + 2) + (8x – 3).

Terms that contain the same variable expression are described as “like terms”

and can be combined via addition or subtraction. In this expression, 5x and

8x are like terms because they both contain x. The constants 2 and –3 are like

terms as well, as they share the same (lack of) variables. Rewrite the expression

by grouping the like terms.

It’s called

the “leading

coefcient” because

you normally write

a polynomial in order

from the highest power

of the variable to the

lowest. That puts the

term with the biggest

exponent out front, and

its coefcient ends

up at the beginning of

the polynomial, in the

“lead” spot.

It’s called

a constant

because with no

variables inside

it, its value never

varies, remaining

constant no matter

what value of

x gets plugged

into the other

terms.

8y and –5y

would be like

terms (because they

both have y). However,

2x

2

and 9x are NOT like

terms. They both have

an x, but those x’s are

raised to different

powers. The variables

of like terms

must match

exactly.

You’re allowed to move the terms

around when you’re adding real numbers,

according to the commutative property. (See

Problem 1.34 for more information.)

Chapter Eleven — Polynomials

The Humongous Book of Algebra Problems

240

(5x + 8x) + (2 – 3)

To combine like terms, add the coefﬁcients and multiply the result by the

shared variable. For example, to add 5x and 8x, add the coefﬁcients (5 + 8 = 13)

and multiply the result by the shared variable (x): 5x + 8x = 13x and 2 – 3 = –1.

(5x + 8x) + (2 – 3) = 13x – 1

11.10 Simplify the expression: (2x

2

+ 3x + 1) + (9x

2

– 6x – 13).

This expression contains three pairs of like terms: 2x

2

and 9x

2

; 3x and –6x; 1

and –13. Rewrite the expression by grouping the like terms.

(2x

2

+ 9x

2

) + (3x – 6x) + (1 – 13)

Combine like terms to simplify the expression.

11x

2

– 3x – 12

11.11 Simplify the expression: .

Before you combine the terms of the expression, distribute and 3 to the

expressions in the adjacent parentheses.

Rewrite the expression by grouping like terms.

Combine like terms, using a common denominator to add the x-terms.

Therefore, .

The distributive

property says that

you can multiply a number

outside parentheses to

an addition or subtraction

problem inside. That means

3(x – 2) = 3(x) + 3(–2) = 3x – 6.

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