QuestionQuestion 4
A factor of $x^{3}+12 x^{2}+41 x+42$ is $x-3$ $x+3$ $x-2$ $x-7$

Studdy Solution

STEP 1

1. We are given the polynomial $x^3 + 12x^2 + 41x + 42$ .
2. We need to determine which of the given options is a factor of the polynomial.
3. The options are $x-3$ , $x+3$ , $x-2$ , and $x-7$ .

STEP 2

1. Use the Factor Theorem to test each option.
2. Substitute the root of each option into the polynomial.
3. Determine which substitution results in the polynomial evaluating to zero.

STEP 3

The Factor Theorem states that if $x - c$ is a factor of a polynomial $f(x)$ , then $f(c) = 0$ .

STEP 4

Test the first option, $x - 3$ . Substitute $x = 3$ into the polynomial:
$f(3) = (3)^3 + 12(3)^2 + 41(3) + 42$

STEP 5

Calculate $f(3)$ :
$f(3) = 27 + 12(9) + 123 + 42$ $f(3) = 27 + 108 + 123 + 42$ $f(3) = 300$
Since $f(3) \neq 0$ , $x - 3$ is not a factor.

STEP 6

Test the second option, $x + 3$ . Substitute $x = -3$ into the polynomial:
$f(-3) = (-3)^3 + 12(-3)^2 + 41(-3) + 42$

STEP 7

Calculate $f(-3)$ :
$f(-3) = -27 + 12(9) - 123 + 42$ $f(-3) = -27 + 108 - 123 + 42$ $f(-3) = 0$
Since $f(-3) = 0$ , $x + 3$ is a factor.

STEP 8

Since $x + 3$ results in the polynomial evaluating to zero, it is a factor of the polynomial.
The factor of the polynomial $x^3 + 12x^2 + 41x + 42$ is:
$\boxed{x + 3}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms