To determine whether [tex]\(x - 8\)[/tex] is a factor of the polynomial [tex]\(x^3 - 3x^2 - 31x - 72\)[/tex], we will use the Factor Theorem. The Factor Theorem states that [tex]\(x - c\)[/tex] is a factor of a polynomial [tex]\(P(x)\)[/tex] if and only if [tex]\(P(c) = 0\)[/tex].
Here, we need to check if [tex]\(x - 8\)[/tex] is a factor, which means we need to evaluate the polynomial at [tex]\(x = 8\)[/tex].
Let [tex]\(P(x) = x^3 - 3x^2 - 31x - 72\)[/tex]. We will substitute [tex]\(x = 8\)[/tex] into [tex]\(P(x)\)[/tex] and check if the result is zero:
[tex]\[ P(8) = 8^3 - 3(8)^2 - 31(8) - 72 \][/tex]
First, calculate each term separately:
- [tex]\(8^3 = 512\)[/tex]
- [tex]\(3(8)^2 = 3 \times 64 = 192\)[/tex]
- [tex]\(31(8) = 248\)[/tex]
- The constant term is [tex]\(-72\)[/tex]
Now, substitute these values into the polynomial:
[tex]\[ P(8) = 512 - 192 - 248 - 72 \][/tex]
Now, add and subtract these values step-by-step:
[tex]\[ 512 - 192 = 320 \][/tex]
[tex]\[ 320 - 248 = 72 \][/tex]
[tex]\[ 72 - 72 = 0 \][/tex]
Since [tex]\(P(8) = 0\)[/tex], [tex]\(x - 8\)[/tex] is indeed a factor of the polynomial [tex]\(x^3 - 3x^2 - 31x - 72\)[/tex].
Therefore, the correct answer is:
A. Yes
