# 3.6 Zeros of polynomial functions  (Page 8/14)

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$f\left(x\right)=2{x}^{3}+3{x}^{2}+x+6;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x+2$

$-2$

$f\left(x\right)=-5{x}^{3}+16{x}^{2}-9;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x-3$

${x}^{3}+3{x}^{2}+4x+12;\text{\hspace{0.17em}}x+3$

$-3$

$4{x}^{3}-7x+3;\text{\hspace{0.17em}}x-1$

$2{x}^{3}+5{x}^{2}-12x-30,\text{​}\text{\hspace{0.17em}}2x+5$

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

${x}^{3}-3{x}^{2}-10x+24=0$

$2{x}^{3}+7{x}^{2}-10x-24=0$

${x}^{3}+2{x}^{2}-9x-18=0$

${x}^{3}+5{x}^{2}-16x-80=0$

${x}^{3}-3{x}^{2}-25x+75=0$

$2{x}^{3}-3{x}^{2}-32x-15=0$

$2{x}^{3}+{x}^{2}-7x-6=0$

$2{x}^{3}-3{x}^{2}-x+1=0$

$3{x}^{3}-{x}^{2}-11x-6=0$

$2{x}^{3}-5{x}^{2}+9x-9=0$

$\frac{3}{2}$

$2{x}^{3}-3{x}^{2}+4x+3=0$

${x}^{4}-2{x}^{3}-7{x}^{2}+8x+12=0$

${x}^{4}+2{x}^{3}-9{x}^{2}-2x+8=0$

$4{x}^{4}+4{x}^{3}-25{x}^{2}-x+6=0$

$2{x}^{4}-3{x}^{3}-15{x}^{2}+32x-12=0$

${x}^{4}+2{x}^{3}-4{x}^{2}-10x-5=0$

$4{x}^{3}-3x+1=0$

$8{x}^{{}^{4}}+26{x}^{3}+39{x}^{2}+26x+6$

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

${x}^{3}+{x}^{2}+x+1=0$

${x}^{3}-8{x}^{2}+25x-26=0$

${x}^{3}+13{x}^{2}+57x+85=0$

$3{x}^{3}-4{x}^{2}+11x+10=0$

${x}^{4}+2{x}^{3}+22{x}^{2}+50x-75=0$

$2{x}^{3}-3{x}^{2}+32x+17=0$

## Graphical

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

$f\left(x\right)={x}^{3}-1$

$f\left(x\right)={x}^{4}-{x}^{2}-1$

1 positive, 1 negative

$f\left(x\right)={x}^{3}-2{x}^{2}-5x+6$

$f\left(x\right)={x}^{3}-2{x}^{2}+x-1$

3 or 1 positive, 0 negative

$f\left(x\right)={x}^{4}+2{x}^{3}-12{x}^{2}+14x-5$

$f\left(x\right)=2{x}^{3}+37{x}^{2}+200x+300$

0 positive, 3 or 1 negative

$f\left(x\right)={x}^{3}-2{x}^{2}-16x+32$

$f\left(x\right)=2{x}^{4}-5{x}^{3}-5{x}^{2}+5x+3$

2 or 0 positive, 2 or 0 negative

$f\left(x\right)=2{x}^{4}-5{x}^{3}-14{x}^{2}+20x+8$

$f\left(x\right)=10{x}^{4}-21{x}^{2}+11$

2 or 0 positive, 2 or 0 negative

## Numeric

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

$f\left(x\right)={x}^{4}+3{x}^{3}-4x+4$

$f\left(x\right)=2{x}^{{}^{3}}+3{x}^{2}-8x+5$

$f\left(x\right)=3{x}^{{}^{3}}+5{x}^{2}-5x+4$

$f\left(x\right)=6{x}^{4}-10{x}^{2}+13x+1$

$f\left(x\right)=4{x}^{5}-10{x}^{4}+8{x}^{3}+{x}^{2}-8$

## 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.

$f\left(x\right)=6{x}^{3}-7{x}^{2}+1$

$f\left(x\right)=4{x}^{3}-4{x}^{2}-13x-5$

$f\left(x\right)=8{x}^{3}-6{x}^{2}-23x+6$

$f\left(x\right)=12{x}^{4}+55{x}^{3}+12{x}^{2}-117x+54$

$f\left(x\right)=16{x}^{4}-24{x}^{3}+{x}^{2}-15x+25$

$\frac{5}{4}$

## Extensions

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

Real roots: –1, 1, 3 and $\text{\hspace{0.17em}}\left(2,f\left(2\right)\right)=\left(2,4\right)$

Real roots: –1 (with multiplicity 2 and 1) and $\text{\hspace{0.17em}}\left(2,f\left(2\right)\right)=\left(2,4\right)$

$f\left(x\right)=\frac{4}{9}\left({x}^{3}+{x}^{2}-x-1\right)$

Real roots: –2, $\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ (with multiplicity 2) and $\text{\hspace{0.17em}}\left(-3,f\left(-3\right)\right)=\left(-3,5\right)$

Real roots: $\text{\hspace{0.17em}}-\frac{1}{2}\text{\hspace{0.17em}}$ , 0, $\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,f\left(-2\right)\right)=\left(-2,6\right)$

$f\left(x\right)=-\frac{1}{5}\left(4{x}^{3}-x\right)$

Real roots: –4, –1, 1, 4 and $\text{\hspace{0.17em}}\left(-2,f\left(-2\right)\right)=\left(-2,10\right)$

## 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.

8 by 4 by 6 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.

5.5 by 4.5 by 3.5 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.

8 by 5 by 3 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 $\text{\hspace{0.17em}}16\pi \text{\hspace{0.17em}}$ cubic meters.

The height is one less than one half the radius. The volume is $\text{\hspace{0.17em}}72\pi \text{\hspace{0.17em}}$ cubic meters.

Radius = 6 meters, Height = 2 meters

The radius and height differ by one meter. The radius is larger and the volume is $\text{\hspace{0.17em}}48\pi \text{\hspace{0.17em}}$ cubic meters.

The radius and height differ by two meters. The height is greater and the volume is $\text{\hspace{0.17em}}28.125\pi \text{\hspace{0.17em}}$ cubic meters.

Radius = 2.5 meters, Height = 4.5 meters

80. The radius is $\text{\hspace{0.17em}}\frac{1}{3}\text{\hspace{0.17em}}$ meter greater than the height. The volume is $\text{\hspace{0.17em}}\frac{98}{9}\pi \text{\hspace{0.17em}}$ cubic meters.

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich