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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
coefcient” 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 coefcient 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 xs are
raised to different
powers. The variables
of like terms
must match
exactly.
Youre allowed to move the terms
around when youre 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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