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Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x)=x^4+10x^3-20x^2-90x+99 What are the real zeros? , fill in the answer box x= Use the real zeros to factor f. type in factored form use radicals as needed f(x)=

          Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.
f(x)=x^4+10x^3-20x^2-90x+99
What are the real zeros? , fill in the answer box 
x=

Use the real zeros to factor f. type in factored form use radicals as needed
f(x)=

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Added by Jason C.

College Algebra

Michael Sullivan 10th Edition

Chapter 5

Instant Answer

Solved by Expert Leslie Deeb

07/27/2023

Step 1

According to the Rational Root Theorem, the potential rational zeros are all the factors of the constant term (99) divided by all the factors of the leading coefficient (1). So, the potential rational zeros are ±1, ±3, ±9, ±11, ±33, ±99. By using synthetic Show more…

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Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x)=x^4+10x^3-20x^2-90x+99 What are the real zeros? , fill in the answer box x= Use the real zeros to factor f. type in factored form use radicals as needed f(x)=

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00:01 Okay, to find the zeros of this, first i have to list what the possible zeros are.

00:11 And to do that, i take all the factors of the constant, and the constant of this function is 99, and divide by all the factors of the leading coefficient.

00:28 And the leading coefficient of this problem is 1.

00:37 So the factors of 99 would be plus or minus 1 and 99, 3 and 33, 9 and 11.

00:54 And they'd all be plus or minus.

00:57 And since i'm dividing them all by 1, that's the only factor of 1.

01:03 I really don't have to worry about the bottom.

01:07 Okay, so what we're going to do is synthetic division with those zeros until we find one that gives us a remainder of zero.

01:16 So let's set this problem up for synthetic division.

01:20 We have 1x to the fourth, 10x to the third, negative 20x to the second, 90x to the first, and a 99.

01:36 And the first number that i'm going to try is positive 1.

01:44 Okay, so in synthetic division, the first thing you do is multiply the number under the line by the number in the box.

01:55 So 1 times 1 is 1.

01:59 Then you add those numbers.

02:02 10 plus 1 is 11.

02:05 Then you multiply again the number under the line times the number in the box.

02:11 11 times 1 is 11.

02:14 And you just keep repeating this process.

02:16 Negative 20 plus 11 is negative 9.

02:20 Negative 9 times 1 is negative 9.

02:25 Negative 90 plus negative 9 is negative 99.

02:31 Negative 99 times 1 is negative 99.

02:36 And that gives me a remainder of zero...

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