Section A.7 Polynomials and Factoring
ΒΆSubsection Polynomials
A polynomial is a sum of terms in which all the exponents on the variables are whole numbers and no variables appear in the denominator or under a radical. The expressionsExample A.47.
Which of the following expressions are polynomials?
\(\pi r^2\)
\(23.4s^6 - 47.9s^4\)
\(\dfrac{2}{3}w^3 - \dfrac{7}{3}w^2 + \dfrac{1}{3}w\)
\(7 + m^{-2}\)
\(\dfrac{x-2}{x+2}\)
\(\sqrt[3]{4y}\)
The first three are all polynomials. In fact, (a) is a monomial, (b) is a binomial, and (c) is a trinomial. The last three are not polynomials. The variable in (d) has a negative exponent, the variable in (e) occurs in the denominator, and the variable in (f) occurs under a radical.
Example A.48.
Give the degree of each polynomial.
\(b^3 - 3b^2 + 3b - 1\)
\(10^{10}\)
\(-4w^3\)
\(s^2 - s^6\)
This is a polynomial in the variable \(b\text{,}\) and because the greatest exponent on \(b\) is \(3\text{,}\) the degree of this polynomial is \(3\text{.}\)
This is a constant polynomial, so its degree is \(0\text{.}\) (The exponent on a constant does not affect the degree.)
This monomial has degree \(3\text{.}\)
This is a binomial of degree \(6\text{.}\)
Example A.49.
Let \(p(x) = -2x^2 + 3x - 1\text{.}\) Evaluate each of the following.
\(p(2)\)
\(p(-1)\)
\(p(t)\)
\(p(t+3)\)
In each case, we replace \(x\) by the given value.
\(p(\alert{2})=-2(\alert{2})^2 + 3(\alert{2}) - 1=-8+6-1=-3\)
\(p(\alert{-1})=-2(\alert{-1})^2 + 3(\alert{-1}) - 1=-2+(-3)-1=-6\)
\(p(\alert{t})=-2(\alert{t})^2 + 3(\alert{t}) - 1=--2t^2+3t-1\)
\(\begin{aligned}[t] \amp \\ p(\alert{t+3})\amp =-2(\alert{t+3})^2 + 3(\alert{t+3}) - 1\\ \amp = -2(t^2+6t+9)+3(t+3)-1\\ \amp = -2t^2-9t-10 \end{aligned}\)
Subsection Products of Polynomials
To multiply polynomials, we use a generalized form of the distributive property:Example A.50.
\(\begin{aligned}[t] \amp\\ 3x(x + y + z) \amp = 3x(x) + 3x(y) + 3x(z)\\ \amp = 3x^2+3xy+3xz \end{aligned}\)
\(\begin{aligned}[t] \amp\\ -2ab^2(3a^2 - ab + 2b^2) \amp = -2ab^2(3a^2) - 2ab^2(-ab) - 2ab^2(2b^2)\\ \amp = -6a^3b^2+2a^2b^3-4ab^4 \end{aligned}\)
Subsection Products of Binomials
Products of binomials occur so frequently that it is worthwhile to learn a shortcut for this type of multiplication. We can use the following scheme to perform the multiplication mentally. (See Figure A.51.)Example A.52.
Subsection Factoring
We sometimes find it useful to write a polynomial as a single term composed of two or more factors. This process is the reverse of multiplication and is called factoring. For example, observe thatSubsection Common Factors
We can factor a common factor from a polynomial by using the distributive property in the formExample A.53.
-
\(\begin{aligned}[t] \\ 18x^2 y - 24xy^2 \amp = 6xy(\text{?} - \text{?}) \\ \amp = 6xy(3x - 4y) \end{aligned}\)
because
\begin{equation*} 6xy(3x - 4y) = 18x^2 y - 24xy^2 \end{equation*} -
\(\begin{aligned}[t] \\ y(x - 2) + z(x - 2) \amp = (x - 2)(\text{?} - \text{?}) \\ \amp = (x - 2)(y + z) \end{aligned}\)
because
\begin{equation*} (x - 2)(y + z) = y(x - 2) + z(x - 2) \end{equation*}
Subsection Opposite of a Binomial
It is often useful to factor β1 from the terms of a binomial.Opposite of a Binomial.
Example A.54.
\(3x - y = -(y - 3x)\)
\(a - 2b = -(2b - a)\)
Subsection Polynomial Division
We can divide one polynomial by a polynomial of lesser degree. The quotient will be the sum of a polynomial and a simpler algebraic fraction. If the divisor is a monomial, we can simply divide the monomial into each term of the numerator.Example A.55.
Divide \(\dfrac{9x^3 - 6x^2 + 4}{3x}\)
Divide \(3x\) into each term of the numerator.
The quotient is the sum of a polynomial, \(3x^2 - 2x\text{,}\) and an algebraic fraction, \(\dfrac{4}{3x}\text{.}\)
Example A.56.
Divide \(\dfrac{2x^2+x-7}{x+3}\)
First write

and divide \(2x^2\) (the first term of the numerator) by \(x\) (the first term of the denominator) to obtain \(2x\text{.}\) (It may be helpful to write down the division: \(\dfrac{2x^2}{2x}=x\text{.}\)) Write \(2x\) above the quotient bar as the first term of the quotient, as shown below.
Next, multiply \(x+3\) by \(2x\) to obtain \(2x^2 + 6x\text{,}\) and subtract this product from \(2x^2 + x - 7\text{:}\)

Repeating the process, divide \(-5x\) by \(x\) to obtain \(-5\text{.}\) Write \(-5\) as the second term of the quotient. Then multiply \(x+3\) by \(-5\) to obtain \(-5x - 15\text{,}\) and subtract:

Because the degree of \(8\) is less than the degree of \(x + 3\text{,}\) the division is finished. The quotient is \(2x - 5\text{,}\) with a remainder of \(8\text{.}\) We write the remainder as a fraction to obtain

Subsection Section Summary
ΒΆSubsubsection Vocabulary
Look up the definitions of new terms in the Glossary.Polynomial
Common factor
Degree
Constant
Trinomial
Monomial
Binomial
Subsubsection SKILLS
Practice each skill in the exercises listed.Identify polynomials: #1β12
Evaluate polynomials: #13β20
Multiply polynomials: #21β42
Factor out a common factor: #43β68
Divide polynomials: #69β80
Exercises Exercises A.7
ΒΆFor Problems 1-8, identify the polynomial as a monomial, a binomial, or a trinomial. Give the degree of the polynomial.
Which of the expressions in Problems 9-12 are not polynomials?
For Problems 13-20, evaluate the polynomial function for the given values of the variable.
13.
P(x)=x3β3x2+x+1
x=2
x=β2
x=2b
14.
P(x)=2x3+x2β3x+4
x=3
x=β3
x=βa
15.
Q(t)=t2+3t+1
t=12
t=β13
t=βw
16.
Q(t)=2t2βt+1
t=14
t=β12
t=3v
17.
R(z)=3z4β2z2+3
z=1.8
z=β2.6
z=kβ1
18.
R(z)=z4+4zβ2
z=2.1
z=β3.1
z=h+2
19.
N(a)=a6βa5
a=β1
a=β2
a=m3
20.
N(a)=a5βa4
a=β1
a=β2
a=q2
For Problems 21-42, write the product as a polynomial and simplify.
21.
4y(xβ2y)
22.
3x(2x+y)
23.
β6x(2x2βx+1)
24.
β2y(y2β3y+2)
25.
a2b(3a2β2abβb)
26.
ab3(βa2b2+4abβ3)
27.
2x2y2(4xy4β2xyβ3x3y2)
28.
5x2y2(3x4y2+3x2yβxy6)
29.
(n+2)(n+8)
30.
(rβ1)(rβ6)
31.
(r+5)(rβ2)
32.
(zβ3)(z+5)
33.
(2z+1)(zβ3)
34.
(3tβ1)(2t+1)
35.
(4r+3s)(2rβs)
36.
(2zβw)(3z+5w)
37.
(2xβ3y)(3xβ2y)
38.
(3a+5b)(3a+4b)
39.
(3tβ4s)(3t+4s)
40.
(2xβ3z)(2x+3z)
41.
(2a2+b2)(a2β3b2)
42.
(s2β5t2)(3s2+2t2)
For Problems 43-60, factor completely. Check your answers by multiplying factors.
43.
4x2z+8xz
44.
3x2y+6xy
45.
3n4β6n3+12n2
46.
2x4β4x2+6x
47.
15r2s+18rs2β3r
48.
2x2y2β3xy+5x2
49.
3m2n4β6m3n3+14m3n2
50.
6x3yβ6xy3+12x2y2
51.
15a4b3c4β12a2b2c5+6a2b3c4
52.
14xy4z3+21x2y3z2β28x3y2z5
53.
a(a+3)+b(a+3)
54.
b(aβ2)+a(aβ2)
55.
y(yβ2)β3x(yβ2)
56.
2x(x+3)βy(x+3)
57.
4(xβ2)2β8x(xβ2)3
58.
6(x+1)β3x(x+1)2
59.
x(xβ5)2βx2(xβ5)3
60.
x2(x+3)3βx(x+3)2
For Problems 61-68, supply the missing factors or terms.
For Problems 69-80, divide.