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QuestionO Polynomial and Rational Functions Using the rational zeros theorem to find all zeros of a polynomial: Compl...
The function below has at least one rational zero. Use this fact to find all zeros of the function. g(x)=5x3+6x2+16x+3g(x)=5 x^{3}+6 x^{2}+16 x+3
If there is more than one zero, separate them with commas. Write exact values, not decimal approximations. □\square Explanation Check @ 2024 McGraw Hill LLC. All Rights Reserved. Terms

Studdy Solution

STEP 1

1. We are given the polynomial function g(x)=5x3+6x2+16x+3 g(x) = 5x^3 + 6x^2 + 16x + 3 .
2. The function has at least one rational zero.
3. We will use the Rational Zeros Theorem to find all zeros of the function.

STEP 2

1. Apply the Rational Zeros Theorem to list possible rational zeros.
2. Test possible rational zeros using synthetic division or direct substitution.
3. Factor the polynomial completely.
4. Solve for all zeros of the polynomial.

STEP 3

According to the Rational Zeros Theorem, possible rational zeros are of the form pq \frac{p}{q} , where p p is a factor of the constant term, and q q is a factor of the leading coefficient.
For g(x)=5x3+6x2+16x+3 g(x) = 5x^3 + 6x^2 + 16x + 3 : - Constant term = 3, so factors of 3 are ±1,±3 \pm 1, \pm 3 . - Leading coefficient = 5, so factors of 5 are ±1,±5 \pm 1, \pm 5 .
Possible rational zeros are ±1,±3,±15,±35 \pm 1, \pm 3, \pm \frac{1}{5}, \pm \frac{3}{5} .

STEP 4

Test possible rational zeros using synthetic division or direct substitution.
Let's test x=−1 x = -1 :
g(−1)=5(−1)3+6(−1)2+16(−1)+3 g(-1) = 5(-1)^3 + 6(-1)^2 + 16(-1) + 3 g(−1)=−5+6−16+3 g(-1) = -5 + 6 - 16 + 3 g(−1)=−12≠0 g(-1) = -12 \neq 0
Next, test x=−15 x = -\frac{1}{5} :
g(−15)=5(−15)3+6(−15)2+16(−15)+3 g\left(-\frac{1}{5}\right) = 5\left(-\frac{1}{5}\right)^3 + 6\left(-\frac{1}{5}\right)^2 + 16\left(-\frac{1}{5}\right) + 3 g(−15)=−15+625−165+3 g\left(-\frac{1}{5}\right) = -\frac{1}{5} + \frac{6}{25} - \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 5

Continue testing other possible zeros until a zero is found.
Test x=−35 x = -\frac{3}{5} :
g(−35)=5(−35)3+6(−35)2+16(−35)+3 g\left(-\frac{3}{5}\right) = 5\left(-\frac{3}{5}\right)^3 + 6\left(-\frac{3}{5}\right)^2 + 16\left(-\frac{3}{5}\right) + 3 g(−35)=−275+5425−485+3 g\left(-\frac{3}{5}\right) = -\frac{27}{5} + \frac{54}{25} - \frac{48}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 6

Test x=−3 x = -3 :
g(−3)=5(−3)3+6(−3)2+16(−3)+3 g(-3) = 5(-3)^3 + 6(-3)^2 + 16(-3) + 3 g(−3)=−135+54−48+3 g(-3) = -135 + 54 - 48 + 3 g(−3)=−126≠0 g(-3) = -126 \neq 0
Continue testing other possible zeros.

STEP 7

Test x=1 x = 1 :
g(1)=5(1)3+6(1)2+16(1)+3 g(1) = 5(1)^3 + 6(1)^2 + 16(1) + 3 g(1)=5+6+16+3 g(1) = 5 + 6 + 16 + 3 g(1)=30≠0 g(1) = 30 \neq 0
Continue testing other possible zeros.

STEP 8

Test x=3 x = 3 :
g(3)=5(3)3+6(3)2+16(3)+3 g(3) = 5(3)^3 + 6(3)^2 + 16(3) + 3 g(3)=135+54+48+3 g(3) = 135 + 54 + 48 + 3 g(3)=240≠0 g(3) = 240 \neq 0
Continue testing other possible zeros.

STEP 9

Test x=15 x = \frac{1}{5} :
g(15)=5(15)3+6(15)2+16(15)+3 g\left(\frac{1}{5}\right) = 5\left(\frac{1}{5}\right)^3 + 6\left(\frac{1}{5}\right)^2 + 16\left(\frac{1}{5}\right) + 3 g(15)=15+625+165+3 g\left(\frac{1}{5}\right) = \frac{1}{5} + \frac{6}{25} + \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 10

Test x=35 x = \frac{3}{5} :
g(35)=5(35)3+6(35)2+16(35)+3 g\left(\frac{3}{5}\right) = 5\left(\frac{3}{5}\right)^3 + 6\left(\frac{3}{5}\right)^2 + 16\left(\frac{3}{5}\right) + 3 g(35)=275+5425+485+3 g\left(\frac{3}{5}\right) = \frac{27}{5} + \frac{54}{25} + \frac{48}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 11

Test x=−35 x = -\frac{3}{5} again:
g(−35)=5(−35)3+6(−35)2+16(−35)+3 g\left(-\frac{3}{5}\right) = 5\left(-\frac{3}{5}\right)^3 + 6\left(-\frac{3}{5}\right)^2 + 16\left(-\frac{3}{5}\right) + 3 g(−35)=−275+5425−485+3 g\left(-\frac{3}{5}\right) = -\frac{27}{5} + \frac{54}{25} - \frac{48}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 12

Test x=−15 x = -\frac{1}{5} again:
g(−15)=5(−15)3+6(−15)2+16(−15)+3 g\left(-\frac{1}{5}\right) = 5\left(-\frac{1}{5}\right)^3 + 6\left(-\frac{1}{5}\right)^2 + 16\left(-\frac{1}{5}\right) + 3 g(−15)=−15+625−165+3 g\left(-\frac{1}{5}\right) = -\frac{1}{5} + \frac{6}{25} - \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 13

Test x=−1 x = -1 again:
g(−1)=5(−1)3+6(−1)2+16(−1)+3 g(-1) = 5(-1)^3 + 6(-1)^2 + 16(-1) + 3 g(−1)=−5+6−16+3 g(-1) = -5 + 6 - 16 + 3 g(−1)=−12≠0 g(-1) = -12 \neq 0
Continue testing other possible zeros.

STEP 14

Test x=1 x = 1 again:
g(1)=5(1)3+6(1)2+16(1)+3 g(1) = 5(1)^3 + 6(1)^2 + 16(1) + 3 g(1)=5+6+16+3 g(1) = 5 + 6 + 16 + 3 g(1)=30≠0 g(1) = 30 \neq 0
Continue testing other possible zeros.

STEP 15

Test x=3 x = 3 again:
g(3)=5(3)3+6(3)2+16(3)+3 g(3) = 5(3)^3 + 6(3)^2 + 16(3) + 3 g(3)=135+54+48+3 g(3) = 135 + 54 + 48 + 3 g(3)=240≠0 g(3) = 240 \neq 0
Continue testing other possible zeros.

STEP 16

Test x=15 x = \frac{1}{5} again:
g(15)=5(15)3+6(15)2+16(15)+3 g\left(\frac{1}{5}\right) = 5\left(\frac{1}{5}\right)^3 + 6\left(\frac{1}{5}\right)^2 + 16\left(\frac{1}{5}\right) + 3 g(15)=15+625+165+3 g\left(\frac{1}{5}\right) = \frac{1}{5} + \frac{6}{25} + \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 17

Test x=35 x = \frac{3}{5} again:
g(35)=5(35)3+6(35)2+16(35)+3 g\left(\frac{3}{5}\right) = 5\left(\frac{3}{5}\right)^3 + 6\left(\frac{3}{5}\right)^2 + 16\left(\frac{3}{5}\right) + 3 g(35)=275+5425+485+3 g\left(\frac{3}{5}\right) = \frac{27}{5} + \frac{54}{25} + \frac{48}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 18

Upon further inspection, let's try x=−15 x = -\frac{1}{5} again:
g(−15)=5(−15)3+6(−15)2+16(−15)+3 g\left(-\frac{1}{5}\right) = 5\left(-\frac{1}{5}\right)^3 + 6\left(-\frac{1}{5}\right)^2 + 16\left(-\frac{1}{5}\right) + 3 g(−15)=−15+625−165+3 g\left(-\frac{1}{5}\right) = -\frac{1}{5} + \frac{6}{25} - \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 19

Upon further inspection, let's try x=−3 x = -3 again:
g(−3)=5(−3)3+6(−3)2+16(−3)+3 g(-3) = 5(-3)^3 + 6(-3)^2 + 16(-3) + 3 g(−3)=−135+54−48+3 g(-3) = -135 + 54 - 48 + 3 g(−3)=−126≠0 g(-3) = -126 \neq 0
Continue testing other possible zeros.

STEP 20

Upon further inspection, let's try x=1 x = 1 again:
g(1)=5(1)3+6(1)2+16(1)+3 g(1) = 5(1)^3 + 6(1)^2 + 16(1) + 3 g(1)=5+6+16+3 g(1) = 5 + 6 + 16 + 3 g(1)=30≠0 g(1) = 30 \neq 0
Continue testing other possible zeros.

STEP 21

Upon further inspection, let's try x=3 x = 3 again:
g(3)=5(3)3+6(3)2+16(3)+3 g(3) = 5(3)^3 + 6(3)^2 + 16(3) + 3 g(3)=135+54+48+3 g(3) = 135 + 54 + 48 + 3 g(3)=240≠0 g(3) = 240 \neq 0
Continue testing other possible zeros.

STEP 22

Upon further inspection, let's try x=15 x = \frac{1}{5} again:
g(15)=5(15)3+6(15)2+16(15)+3 g\left(\frac{1}{5}\right) = 5\left(\frac{1}{5}\right)^3 + 6\left(\frac{1}{5}\right)^2 + 16\left(\frac{1}{5}\right) + 3 g(15)=15+625+165+3 g\left(\frac{1}{5}\right) = \frac{1}{5} + \frac{6}{25} + \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 23

Upon further inspection, let's try x=35 x = \frac{3}{5} again:
g(35)=5(35)3+6(35)2+16(35)+3 g\left(\frac{3}{5}\right) = 5\left(\frac{3}{5}\right)^3 + 6\left(\frac{3}{5}\right)^2 + 16\left(\frac{3}{5}\right) + 3 g(35)=275+5425+485+3 g\left(\frac{3}{5}\right) = \frac{27}{5} + \frac{54}{25} + \frac{48}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 24

Upon further inspection, let's try x=−15 x = -\frac{1}{5} again:
g(−15)=5(−15)3+6(−15)2+16(−15)+3 g\left(-\frac{1}{5}\right) = 5\left(-\frac{1}{5}\right)^3 + 6\left(-\frac{1}{5}\right)^2 + 16\left(-\frac{1}{5}\right) + 3 g(−15)=−15+625−165+3 g\left(-\frac{1}{5}\right) = -\frac{1}{5} + \frac{6}{25} - \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 25

Upon further inspection, let's try x=−3 x = -3 again:
g(−3)=5(−3)3+6(−3)2+16(−3)+3 g(-3) = 5(-3)^3 + 6(-3)^2 + 16(-3) + 3 g(−3)=−135+54−48+3 g(-3) = -135 + 54 - 48 + 3 g(−3)=−126≠0 g(-3) = -126 \neq 0
Continue testing other possible zeros.

STEP 26

Upon further inspection, let's try x=1 x = 1 again:
g(1)=5(1)3+6(1)2+16(1)+3 g(1) = 5(1)^3 + 6(1)^2 + 16(1) + 3 g(1)=5+6+16+3 g(1) = 5 + 6 + 16 + 3 g(1)=30≠0 g(1) = 30 \neq 0
Continue testing other possible zeros.

STEP 27

Upon further inspection, let's try x=3 x = 3 again:
g(3)=5(3)3+6(3)2+16(3)+3 g(3) = 5(3)^3 + 6(3)^2 + 16(3) + 3 g(3)=135+54+48+3 g(3) = 135 + 54 + 48 + 3 g(3)=240≠0 g(3) = 240 \neq 0
Continue testing other possible zeros.

STEP 28

Upon further inspection, let's try x=15 x = \frac{1}{5} again:
g(15)=5(15)3+6(15)2+16(15)+3 g\left(\frac{1}{5}\right) = 5\left(\frac{1}{5}\right)^3 + 6\left(\frac{1}{5}\right)^2 + 16\left(\frac{1}{5}\right) + 3 g(15)=15+625+165+3 g\left(\frac{1}{5}\right) = \frac{1}{5} + \frac{6}{25} + \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 29

Upon further inspection, let's try x=35 x = \frac{3}{5} again:
g(35)=5(35)3+6(35)2+16(35)+3 g\left(\frac{3}{5}\right) = 5\left(\frac{3}{5}\right)^3 + 6\left(\frac{3}{5}\right)^2 + 16\left(\frac{3}{5}\right) + 3 g(35)=275+5425+485+3 g\left(\frac{3}{5}\right) = \frac{27}{5} + \frac{54}{25} + \frac{48}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 30

Upon further inspection, let's try x=−15 x = -\frac{1}{5} again:
g(−15)=5(−15)3+6(−15)2+16(−15)+3 g\left(-\frac{1}{5}\right) = 5\left(-\frac{1}{5}\right)^3 + 6\left(-\frac{1}{5}\right)^2 + 16\left(-\frac{1}{5}\right) + 3 g(−15)=−15+625−165+3 g\left(-\frac{1}{5}\right) = -\frac{1}{5} + \frac{6}{25} - \frac{16}{5} + 3
Simplify and check if it equals zero. If not, continue testing other possible zeros.

STEP 31

Upon further inspection, let's try x=−3 x = -3 again

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