5.5.2: Homework

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Reading Questions

What are the four main components in the Division Algorithm for polynomials: $f(x) = d(x)q(x) + r(x)$? Define each term.
When performing long division of polynomials, how do you know when the division process is complete? What condition must the remainder satisfy?
If the remainder of a polynomial division is $0$, what does this imply about the relationship between the divisor, the dividend, and the quotient?
Describe the first step in setting up polynomial long division (e.g., dividing $2x^3-3x^2+4x+5$ by $x+2$).
In polynomial long division, after multiplying the current term of the quotient by the entire divisor, what is the next arithmetic operation performed?
What is synthetic division, and for what special case of divisor is it typically used?
In synthetic division, if you are dividing by $x-k$, what value representing the divisor do you write outside the division symbol? What if you are dividing by $x+k$?
Describe the process of bringing down the leading coefficient and the subsequent multiply-add steps in synthetic division.
How do you interpret the bottom row of numbers in a completed synthetic division problem to find the quotient and remainder?
If a polynomial dividend is missing a term (e.g., no $x^2$ term in a cubic polynomial), how should this be handled when setting up either long division or synthetic division?
How can you check your work after performing polynomial division?
Can polynomial division be used to find unknown dimensions of a geometric solid if its volume and some other dimensions are given as polynomials?
If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?
If a polynomial of degree $n$ is divided by a binomial of degree 1, what is the degree of the quotient?

Homework

Algebraic

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

$( x^2 +5x−1 ) \div ( x−1 )$
$( 2 x^2 −9x−5 ) \div ( x−5 )$
$ ( 3 x^2 + 23x+14 ) \div ( x+7 ) $
$ ( 4 x^2 − 10x+6 ) \div ( 4x+2 ) $
$ ( 6 x^2 −25x−25 ) \div ( 6x+5 ) $
$( − x^2 −1 ) \div ( x+1 )$
$( 2 x^2 −3x+2 ) \div ( x+2 )$
$( x^3 −126 ) \div ( x−5 )$
$( 3 x^2 −5x+4 ) \div ( 3x+1 )$
$( x^3 −3 x^2 +5x−6 ) \div ( x−2 )$
$( 2 x^3 +3 x^2 −4x+15 ) \div ( x+3 )$

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

$( 3 x^3 −2 x^2 +x−4 ) \div ( x+3 )$
$( 2 x^3 −6 x^2 −7x+6 ) \div (x−4)$
$( 6 x^3 −10 x^2 −7x−15 ) \div (x+1)$
$( 4 x^3 −12 x^2 −5x−1 ) \div (2x+1)$
$ ( 9 x^3 −9 x^2 + 18x+5 ) \div (3x−1) $
$( 3 x^3 −2 x^2 +x−4 ) \div ( x+3 )$
$( −6 x^3 + x^2 −4 ) \div ( 2x−3 )$
$ ( 2 x^3 +7 x^2 − 13x−3 ) \div ( 2x−3 ) $
$( 3 x^3 −5 x^2 +2x+3 ) \div (x+2)$
$( 4 x^3 −5 x^2 +13 ) \div (x+4)$
$( x^3 −3x+2 ) \div ( x+2 )$
$ ( x^3 −21 x^2 + 147x−343 ) \div ( x−7 ) $
$ ( x^3 −15 x^2 + 75x−125 ) \div ( x−5 ) $
$( 9 x^3 −x+2 ) \div ( 3x−1 )$
$( 6 x^3 − x^2 +5x+2 ) \div ( 3x+1 )$
$( x^4 + x^3 −3 x^2 −2x+1 ) \div ( x+1 )$
$( x^4 −3 x^2 +1 ) \div ( x−1 )$
$( x^4 +2 x^3 −3 x^2 +2x+6 ) \div ( x+3 )$
$ ( x^4 −10 x^3 +37 x^2 − 60x+36 ) \div ( x−2 ) $
$ ( x^4 −8 x^3 +24 x^2 − 32x+16 ) \div ( x−2 ) $
$ ( x^4 +5 x^3 −3 x^2 − 13x+10 ) \div ( x+5 ) $
$ ( x^4 −12 x^3 +54 x^2 − 108x+81 ) \div ( x−3 ) $
$( 4 x^4 −2 x^3 −4x+2 ) \div ( 2x−1 )$
$( 4 x^4 +2 x^3 −4 x^2 +2x+2 ) \div ( 2x+1 )$

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.

$x−2$, $4 x^3 −3 x^2 −8x+4$
$x−2$, $3 x^4 −6 x^3 −5x+10$
$x+3$, $−4 x^3 +5 x^2 +8$
$x−2$, $4 x^4 −15 x^2 −4$
$x− \frac{1}{2}$, $2 x^4 − x^3 +2x−1$
$x+ \frac{1}{3}$, $3 x^4 + x^3 −3x+1$

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 $x^2 +2x+4$
$Graph of a polynomial that has a x-intercept at 1.$
Factor is $x^2 +2x+5$
$Graph of a polynomial that has a x-intercept at 2.$
Factor is $x^2 +x+1$
$Graph of a polynomial that has a x-intercept at 5.$
Factor is $x^2 +2x+2$
$Graph of a polynomial that has a x-intercept at -3.$

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

$\frac{4 x^3 −33}{x−2}$
$\frac{2 x^3 +25}{x+3}$
$\frac{3 x^3 +2x−5}{x−1}$
$\frac{−4 x^3 − x^2 −12}{x+4}$
$\frac{x^4 −22}{x+2}$

Technology

For the following exercises, use technology to answer the questions.

Consider $\frac{x^k −1}{x−1}$ with $k=1$, $2$, and $3$. What do you expect the result to be if $k=4$?
Consider $\frac{x^k +1}{x+1}$ for $k=1$, $3$, and $5$. What do you expect the result to be if $k=7$?
Consider $\frac{x^4 − k^4}{x−k}$ for $k=1$, $2$, and $3$. What do you expect the result to be if $k=4$?
Consider $\frac{x^k}{x+1}$ with $k=1$, $2$, and $3$. What do you expect the result to be if $k=4$?
Consider $\frac{x^k}{x−1}$ with $k=1$, $2$, and $3$. 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.

$\frac{x+1}{x−i}$
$\frac{x^2 +1}{x−i}$
$\frac{x+1}{x+i}$
$\frac{x^2 +1}{x+i}$
$\frac{x^3 +1}{x−i}$

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 $2 x^2 +9x−5$.
Length is $2x+5$, area is $4 x^3 +10 x^2 +6x+15$
Length is $3x–4$, area is $6 x^4 −8 x^3 +9 x^2 −9x−4$

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 $12 x^3 +20 x^2 −21x−36$, length is $2x+3$, width is $3x−4$.
Volume is $18 x^3 −21 x^2 −40x+48$, length is $3x–4$, width is $3x–4$.
Volume is $10 x^3 +27 x^2 +2x−24$, length is $5x–4$, width is $2x+3$.
Volume is $10 x^3 +30 x^2 −8x−24$, 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 $\pi (25 x^3 −65 x^2 −29x−3)$, radius is $5x+1$.
Volume is $\pi (4 x^3 +12 x^2 −15x−50)$, radius is $2x+5$.
Volume is $\pi (3 x^4 +24 x^3 +46 x^2 −16x−32)$, radius is $x+4$.

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