3.4 Dividing polynomials (Page 4/6)

<< Chapter < Page

Essential precalculus, part 1

Page 4 / 6

Chapter >> Page >

Section exercises

Verbal

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

The binomial is a factor of the polynomial.

If a polynomial of degree $n$ is divided by a binomial of degree 1, what is the degree of the quotient?

Algebraic

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(x^{2} + 5 x - 1) \div (x - 1)$

$x + 6 + \frac{5}{x - 1}, quotient: x + 6, remainder: 5$

$(2 x^{2} - 9 x - 5) \div (x - 5)$

$(3 x^{2} + 23 x + 14) \div (x + 7)$

$3 x + 2, quotient: 3 x + 2, remainder: 0$

$(4 x^{2} - 10 x + 6) \div (4 x + 2)$

$(6 x^{2} - 25 x - 25) \div (6 x + 5)$

$x - 5, quotient: x - 5, remainder: 0$

$(- x^{2} - 1) \div (x + 1)$

$(2 x^{2} - 3 x + 2) \div (x + 2)$

$2 x - 7 + \frac{16}{x + 2}, quotient: 2 x - 7, remainder: 16$

$(x^{3} - 126) \div (x - 5)$

$(3 x^{2} - 5 x + 4) \div (3 x + 1)$

$x - 2 + \frac{6}{3 x + 1}, quotient: x - 2, remainder: 6$

$(x^{3} - 3 x^{2} + 5 x - 6) \div (x - 2)$

$(2 x^{3} + 3 x^{2} - 4 x + 15) \div (x + 3)$

$2 x^{2} - 3 x + 5, quotient: 2 x^{2} - 3 x + 5, remainder: 0$

For the following exercises, use synthetic division to find the quotient.

$(3 x^{3} - 2 x^{2} + x - 4) \div (x + 3)$

$(2 x^{3} - 6 x^{2} - 7 x + 6) \div (x - 4)$

$2 x^{2} + 2 x + 1 + \frac{10}{x - 4}$

$(6 x^{3} - 10 x^{2} - 7 x - 15) \div (x + 1)$

$(4 x^{3} - 12 x^{2} - 5 x - 1) \div (2 x + 1)$

$2 x^{2} - 7 x + 1 - \frac{2}{2 x + 1}$

$(9 x^{3} - 9 x^{2} + 18 x + 5) \div (3 x - 1)$

$(3 x^{3} - 2 x^{2} + x - 4) \div (x + 3)$

$3 x^{2} - 11 x + 34 - \frac{106}{x + 3}$

$(- 6 x^{3} + x^{2} - 4) \div (2 x - 3)$

$(2 x^{3} + 7 x^{2} - 13 x - 3) \div (2 x - 3)$

$x^{2} + 5 x + 1$

$(3 x^{3} - 5 x^{2} + 2 x + 3) \div (x + 2)$

$(4 x^{3} - 5 x^{2} + 13) \div (x + 4)$

$4 x^{2} - 21 x + 84 - \frac{323}{x + 4}$

$(x^{3} - 3 x + 2) \div (x + 2)$

$(x^{3} - 21 x^{2} + 147 x - 343) \div (x - 7)$

$x^{2} - 14 x + 49$

$(x^{3} - 15 x^{2} + 75 x - 125) \div (x - 5)$

$(9 x^{3} - x + 2) \div (3 x - 1)$

$3 x^{2} + x + \frac{2}{3 x - 1}$

$(6 x^{3} - x^{2} + 5 x + 2) \div (3 x + 1)$

$(x^{4} + x^{3} - 3 x^{2} - 2 x + 1) \div (x + 1)$

$x^{3} - 3 x + 1$

$(x^{4} - 3 x^{2} + 1) \div (x - 1)$

$(x^{4} + 2 x^{3} - 3 x^{2} + 2 x + 6) \div (x + 3)$

$x^{3} - x^{2} + 2$

$(x^{4} - 10 x^{3} + 37 x^{2} - 60 x + 36) \div (x - 2)$

$(x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16) \div (x - 2)$

$x^{3} - 6 x^{2} + 12 x - 8$

$(x^{4} + 5 x^{3} - 3 x^{2} - 13 x + 10) \div (x + 5)$

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) \div (x - 3)$

$x^{3} - 9 x^{2} + 27 x - 27$

$(4 x^{4} - 2 x^{3} - 4 x + 2) \div (2 x - 1)$

$(4 x^{4} + 2 x^{3} - 4 x^{2} + 2 x + 2) \div (2 x + 1)$

$2 x^{3} - 2 x + 2$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x - 2, 4 x^{3} - 3 x^{2} - 8 x + 4$

$x - 2, 3 x^{4} - 6 x^{3} - 5 x + 10$

Yes $(x - 2) (3 x^{3} - 5)$

$x + 3, - 4 x^{3} + 5 x^{2} + 8$

$x - 2, 4 x^{4} - 15 x^{2} - 4$

Yes $(x - 2) (4 x^{3} + 8 x^{2} + x + 2)$

$x - \frac{1}{2}, 2 x^{4} - x^{3} + 2 x - 1$

$x + \frac{1}{3}, 3 x^{4} + x^{3} - 3 x + 1$

Graphical

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is $x^{2} - x + 3$

Graph of a polynomial that has a x-intercept at -1.

Factor is $(x^{2} + 2 x + 4)$

$(x - 1) (x^{2} + 2 x + 4)$

Factor is $x^{2} + 2 x + 5$

Graph of a polynomial that has a x-intercept at 2.

Factor is $x^{2} + x + 1$

Graph of a polynomial that has a x-intercept at 5.

$(x - 5) (x^{2} + x + 1)$

Factor is $x^{2} + 2 x + 2$

Graph of a polynomial that has a x-intercept at -3.

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{4 x^{3} - 33}{x - 2}$

$Quotient: 4 x^{2} + 8 x + 16, remainder: - 1$

$\frac{2 x^{3} + 25}{x + 3}$

$\frac{3 x^{3} + 2 x - 5}{x - 1}$

$Quotient: 3 x^{2} + 3 x + 5, remainder: 0$

$\frac{- 4 x^{3} - x^{2} - 12}{x + 4}$

$\frac{x^{4} - 22}{x + 2}$

$Quotient: x^{3} - 2 x^{2} + 4 x - 8, remainder: - 6$

Technology

For the following exercises, use a calculator with CAS to answer the questions.

Consider $\frac{x^{k} - 1}{x - 1}$ with $k = 1, 2, 3.$ What do you expect the result to be if $k = 4 ?$

Consider $\frac{x^{k} + 1}{x + 1}$ for $k = 1, 3, 5.$ What do you expect the result to be if $k = 7 ?$

$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$

Consider $\frac{x^{4} - k^{4}}{x - k}$ for $k = 1, 2, 3.$ What do you expect the result to be if $k = 4 ?$

Consider $\frac{x^{k}}{x + 1}$ with $k = 1, 2, 3.$ What do you expect the result to be if $k = 4 ?$

$x^{3} - x^{2} + x - 1 + \frac{1}{x + 1}$

Consider $\frac{x^{k}}{x - 1}$ with $k = 1, 2, 3.$ What do you expect the result to be if $k = 4 ?$

Extensions

For the following exercises, use synthetic division to determine the quotient involving a complex number.

$\frac{x + 1}{x - i}$

$1 + \frac{1 + i}{x - i}$

$\frac{x^{2} + 1}{x - i}$

$\frac{x + 1}{x + i}$

$1 + \frac{1 - i}{x + i}$

$\frac{x^{2} + 1}{x + i}$

$\frac{x^{3} + 1}{x - i}$

$x^{2} - i x - 1 + \frac{1 - i}{x - i}$

Real-world applications

For the following exercises, use the given length and area of a rectangle to express the width algebraically.

Length is $x + 5,$ area is $2 x^{2} + 9 x - 5.$

Length is $2 x + 5,$ area is $4 x^{3} + 10 x^{2} + 6 x + 15$

$2 x^{2} + 3$

Length is $3 x - 4,$ area is $6 x^{4} - 8 x^{3} + 9 x^{2} - 9 x - 4$

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

Volume is $12 x^{3} + 20 x^{2} - 21 x - 36,$ length is $2 x + 3,$ width is $3 x - 4.$

$2 x + 3$

Volume is $18 x^{3} - 21 x^{2} - 40 x + 48,$ length is $3 x - 4,$ width is $3 x - 4.$

Volume is $10 x^{3} + 27 x^{2} + 2 x - 24,$ length is $5 x - 4,$ width is $2 x + 3.$

$x + 2$

Volume is $10 x^{3} + 30 x^{2} - 8 x - 24,$ length is $2,$ width is $x + 3.$

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

Volume is $π (25 x^{3} - 65 x^{2} - 29 x - 3),$ radius is $5 x + 1.$

$x - 3$

Volume is $π (4 x^{3} + 12 x^{2} - 15 x - 50),$ radius is $2 x + 5.$

Volume is $π (3 x^{4} + 24 x^{3} + 46 x^{2} - 16 x - 32),$ radius is $x + 4.$

$3 x^{2} - 2$

<< Chapter < Page Page > Chapter >>

Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Essential precalculus, part 1' conversation and receive update notifications?

Ask

	10 Sociology 10 Global Inequality MCQ By OpenStax Start Quiz
	33 Biology 33 The Animal Body Basic Form Function By OpenStax Start Quiz
	1 AP 01 Human Body Anatomy Physiology Essay By OpenStax Start Flashcards
	7 AP Key Terms 07 Axial Skeleton By OpenStax Start Key Terms
	19 AP 19 Cardiovascular System Heart MCQ By OpenStax Start Quiz
	43 Biology 43 Animal Reproduction Development MCQ By OpenStax Start Quiz
	12 Dr Dowers Endocrinology By Brooke Delaney Start Exam
	28 AP 28 Development Inheritance MCQ By OpenStax Start Quiz
	3 AP 03 Cellular Level of Organization Essay By OpenStax Start Flashcards
	Interventionism Mixed Economy By Robert Murphy Start Test