Question

Is (x + 7) a factor of f(x) = x^3 − 3x^2 + 2^x − 8? Use either the remainder theorem or the factor theorem to explain your reasoning.

Answer 1

is not a factor

Explanation:

Step-by-step explanation

We know that,

The factor theorem is a theorem that links the factors and the roots of a polynomial.

The theorem is as follows:

A polynomial f(x) has a factor (x−p) if and only if f(p)=0.

Consider,

f(x)=x

3

−3x

2

+2x−8

& (x+7) =(x−(−7))

Here,

p=−7

Now, lets check:

f(−7)=(−7)

3

−3(−7)

2

+2(−7)−8

f(−7)=(−343)−3(49)−14−8

f(−7)=−343−147−14−8

f(−7)=−512 , which is not equal to 0 .

So, According to the Factor theorem, we got

(x+7) is not a factor of f(x)=x

3

−3x

2

+2x−8.

Answer 2

Not a factor

Explanation:

We can use Factor Theorem to answer this question. According to this theorem, in order to find if (x - a) is a factor of a polynomial f(x), calculate f(a). If f(a) comes out to be equal to zero, this will mean that (x-a) is factor of f(x).

Here, the expression we have is (x + 7), so we need to find f(-7) in order to check if (x+7) is a factor of f(x) or not

`f(x)=x^(3)-3x^(2)+2x-8`

Substituting x = -7, we get:

`f(-7)=(-7)^(3)-3(-7)^(2)+2(-7)-8\\\\ f(-7)=-512`

Since f(-7) ≠ 0, (x + 7) is not a factor of the polynomial f(x)