Question

Is (x + 5) a factor of f(x) = x^3 – 4x^2 + 3x + 7? Use either the remainder theorem or the factor theorem to explain your reasoning.

Answer 1

ANSWER

Not a factor

EXPLANATION

The given polynomial has equation:

$f(x) = {x}^{3} - 4 {x}^{2} + 3x + 7$

According to the remainder theorem, if (x+5) is a remainder, then f(-5)=0.

Let us substitute x=-5 into the function and evaluate to see if it will simplify to zero.

$f( - 5) = {( - 5)}^{3} - 4 {( - 5)}^{2} + 3( - 5)+ 7$

$f( - 5) = - 125 - 100 - 15+ 7$

$f( - 5) = - 233$

Since the remainder is not zero, x+5 is not a factor of

$f(x) = {x}^{3} - 4 {x}^{2} + 3x + 7$

Answer 2

Answer:

(x+5) is NOT a factor of given function f(x).

Step-by-step explanation:

Given function is $f\left(x\right)=x^3-4x^2+3x+7$.

Now we need to determine if (x+5) is a factor of given function f(x) or not using either the remainder theorem or the factor theorem.

I'm going to use factor theorem which says that "If (x-c) is a factor of f(x) then f(c)=0"

so for (x+5) we have c=-5

Now we need to check if f(-5)=0 or not

$f\left(x\right)=x^3-4x^2+3x+7$

$f\left(-5\right)=(-5)^3-4(-5)^2+3(-5)+7$

$f\left(-5\right)=125-4(25)-15+7$

$f\left(-5\right)=125-100-15+7$

$f\left(-5\right)=-233$

which is not equal to 0.

Hence (x+5) is NOT a factor of given function f(x).