Math  /  Algebra

QuestionA factor of x3+15x2+71x+105x^{3}+15 x^{2}+71 x+105 is x+5x+5 x5x-5 x7x-7 x3x-3

Studdy Solution

STEP 1

1. We are given the polynomial f(x)=x3+15x2+71x+105 f(x) = x^3 + 15x^2 + 71x + 105 .
2. We need to determine which of the given options is a factor of the polynomial.

STEP 2

1. Use the Factor Theorem to test each option.
2. Substitute each option into the polynomial.
3. Determine which substitution results in the polynomial equaling zero.
4. Conclude which option is a factor.

STEP 3

The Factor Theorem states that xc x - c is a factor of f(x) f(x) if and only if f(c)=0 f(c) = 0 .

STEP 4

Test the first option x+5 x + 5 . Substitute x=5 x = -5 into f(x) f(x) :
f(5)=(5)3+15(5)2+71(5)+105 f(-5) = (-5)^3 + 15(-5)^2 + 71(-5) + 105

STEP 5

Calculate f(5) f(-5) :
f(5)=125+15(25)355+105 f(-5) = -125 + 15(25) - 355 + 105 f(5)=125+375355+105 f(-5) = -125 + 375 - 355 + 105 f(5)=0 f(-5) = 0
Since f(5)=0 f(-5) = 0 , x+5 x + 5 is a factor.

STEP 6

Since we found that f(5)=0 f(-5) = 0 , we do not need to test the other options. We have determined that x+5 x + 5 is a factor.
The factor of the polynomial is:
x+5 \boxed{x + 5}

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