# 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.

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?
a²=4