5.6.2: Homework

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Reading Questions

What does the Remainder Theorem state about dividing a polynomial $f(x)$ by $x-k$? How can it be used to evaluate $f(k)$?
Explain the Factor Theorem. How does it relate the zeros of a polynomial to its factors?
What is the Rational Zeros Theorem used for? What conditions must the coefficients of the polynomial satisfy to use this theorem?
If $\frac{p}{q}$ is a possible rational zero of $f(x) = a_n x^n + \dots + a_0$, what do $p$ and $q$ represent?
According to the Fundamental Theorem of Algebra, how many complex zeros does a polynomial of degree $n > 0$ have (counting multiplicities)?
What does the Complete Factorization Theorem say about factoring a polynomial of degree $n > 0$?
What is the Complex Conjugate Theorem? If $2+3i$ is a zero of a polynomial with real coefficients, what other zero must it have?
If you are asked to find a polynomial of a certain degree with given zeros, what is the general approach using its factors? What role does the leading coefficient $a$ play?
When using synthetic division to test for zeros from the list generated by the Rational Zeros Theorem, what remainder value indicates that a number is indeed a zero?
If you find a zero $k$ of a polynomial $f(x)$, how can you use this information to find other zeros? (Hint: consider the quotient polynomial).
Describe a use for the Remainder Theorem.
Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.
What is the difference between rational and real zeros?
If synthetic division reveals a zero, why should we try that value again as a possible solution?

Homework

Algebraic

For the following exercises, use the Remainder Theorem to find the remainder.

$\left(x^4-9 x^2+14\right) \div(x-2)$
$\left(3 x^3-2 x^2+x-4\right) \div(x+3)$
$\left(x^4+5 x^3-4 x-17\right) \div(x+1)$
$\left(-3 x^2+6 x+24\right) \div(x-4)$
$\left(5 x^5-4 x^4+3 x^3-2 x^2+x-1\right) \div(x+6)$
$\left(x^4-1\right) \div(x-4)$
$\left(3 x^3+4 x^2-8 x+2\right) \div(x-3)$
$\left(4 x^3+5 x^2-2 x+7\right) \div(x+2)$

For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.

$f(x)=2 x^3-9 x^2+13 x-6; \quad x-1$
$f(x)=2 x^3+x^2-5 x+2; \quad x+2$
$f(x)=3 x^3+x^2-20 x+12; \quad x+3$
$f(x)=2 x^3+3 x^2+x+6; \quad x+2$
$f(x)=-5 x^3+16 x^2-9; \quad x-3$
$x^3+3 x^2+4 x+12; \quad x+3$
$4 x^3-7 x+3; \quad x-1$
$2 x^3+5 x^2-12 x-30; \quad 2 x+5$

For the following exercises, use the Rational Zeros Theorem to find all real zeros.

$x^3-3 x^2-10 x+24=0$
$2 x^3+7 x^2-10 x-24=0$
$x^3+2 x^2-9 x-18=0$
$x^3+5 x^2-16 x-80=0$
$x^3-3 x^2-25 x+75=0$
$2 x^3-3 x^2-32 x-15=0$
$2 x^3+x^2-7 x-6=0$
$2 x^3-3 x^2-x+1=0$
$3 x^3-x^2-11 x-6=0$
$2 x^3-5 x^2+9 x-9=0$
$2 x^3-3 x^2+4 x+3=0$
$x^4-2 x^3-7 x^2+8 x+12=0$
$x^4+2 x^3-9 x^2-2 x+8=0$
$4 x^4+4 x^3-25 x^2-x+6=0$
$2 x^4-3 x^3-15 x^2+32 x-12=0$
$x^4+2 x^3-4 x^2-10 x-5=0$
$4 x^3-3 x+1=0$
$8 x^4+26 x^3+39 x^2+26 x+6$

For the following exercises, find all complex solutions (real and non-real).

$x^3+x^2+x+1=0$
$x^3-8 x^2+25 x-26=0$
$x^3+13 x^2+57 x+85=0$
$3 x^3-4 x^2+11 x+10=0$
$x^4+2 x^3+22 x^2+50 x-75=0$
$2 x^3-3 x^2+32 x+17=0$

Numeric

For the following exercises, list all possible rational zeros for the functions.

$f(x)=x^4+3 x^3-4 x+4$
$f(x)=2 x^3+3 x^2-8 x+5$
$f(x)=3 x^3+5 x^2-5 x+4$
$f(x)=6 x^4-10 x^2+13 x+1$
$f(x)=4 x^5-10 x^4+8 x^3+x^2-8$

For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

$Graph of a positive odd-degree polynomial with zeros at x=-2, 1, and 3.$
$Graph of a negative odd-degree polynomial with zeros at x=-3, 1, and 3.$
$Graph of a negative odd-degree polynomial with zeros at x=-1, and 2.$
$Graph of a positive odd-degree polynomial with zeros at x=-2, and 3.$
$Graph of a negative even-degree polynomial with zeros at x=-3, -2, 3, and 4.$

For the following exercises, use the graphs to write a polynomial function of least degree.

$Graph of a positive odd-degree polynomial with zeros at x = negative 2/3, 1/2, and 4/3 and y = 8.$
$Graph of a positive odd-degree polynomial with zeros at x=--200, and 500 and y=50000000.$
$Graph of a positive odd-degree polynomial with zeros at x=--300, and 100 and y=-90000.$

Technology

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

$f(x)=6 x^3-7 x^2+1$
$f(x)=4 x^3-4 x^2-13 x-5$
$f(x)=8 x^3-6 x^2-23 x+6$
$f(x)=12 x^4+55 x^3+12 x^2-117 x+54$
$f(x)=16 x^4-24 x^3+x^2-15 x+25$

Extensions

For the following exercises, use the given information about the polynomial graph to write the equation.

Degree 3. Zeros at $x=–2$, $x=1$, and $x=3$. $ y$-intercept at $(0,–4)$.
Degree 3. Zeros at $x=-5$, $x=–2$, and $x=1$. $ y$-intercept at $(0,6)$.
Degree 5. Roots of multiplicity 2 at $x=3$ and $x=1$ , and a root of multiplicity 1 at $x=–3$. $ y$-intercept at $(0,9)$.
Degree 4. Root of multiplicity 2 at $x=4$, and a roots of multiplicity 1 at $x=1$ and $x=–2$. $ y$-intercept at $(0,-3)$.
Degree 5. Double zero at $x=1$, and triple zero at $x=3$. Passes through the point $(2,15)$.
Degree 3. Zeros at $x=4$, $x=3$, and $x=2$. $ y$-intercept at $( 0,−24 )$.
Degree 3. Zeros at $x=−3$, $x=−2$ and $x=1$. $ y$-intercept at $(0,12)$.
Degree 5. Roots of multiplicity 2 at $x=−3$ and $x=2$ and a root of multiplicity 1 at $x=−2$. $ y$-intercept at $( 0,4 )$.
Degree 4. Roots of multiplicity 2 at $x= \frac{1}{2} $ and roots of multiplicity 1 at $x=6$ and $x=−2$. $ y$-intercept at $( 0,18 )$.
Double zero at $x=−3$ and triple zero at $x=0$. Passes through the point $ (1,32)$.

For the following exercises, construct a polynomial function of least degree possible using the given information.

Real roots: –1, 1, 3 and $(2, f(2))=(2,4)$
Real roots: –1, 1 (with multiplicity 2 and 1) and $(2, f(2))=(2,4)$
Real roots: $-2, \frac{1}{2}$ (with multiplicity 2) and $(-3, f(-3))=(-3,5)$
Real roots: $-\frac{1}{2}, 0, \frac{1}{2}$ and $(-2, f(-2))=(-2,6)$
Real roots: –4, –1, 1, 4 and $(-2, f(-2))=(-2,10)$

Real-World Applications

For the following exercises, find the dimensions of the box described.

The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.
The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.
The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.
The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.
The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

For the following exercises, find the dimensions of the right circular cylinder described.

The radius is 3 inches more than the height. The volume is $16 \pi$ cubic inches.
The height is one less than one half the radius. The volume is $72 \pi$ cubic meters.
The radius and height differ by one meter. The radius is larger and the volume is $48 \pi$ cubic meters.
The radius and height differ by two meters. The height is greater and the volume is $28.125 \pi$ cubic meters.
The radius is $\frac{1}{3}$ meter greater than the height. The volume is $\frac{98}{9} \pi$ cubic meters.

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