To determine whether [tex]\(x + 1\)[/tex] is a factor of the polynomial [tex]\(P(x) = -2x^3 - 3x^2 - 4x + 6\)[/tex] using the Factor Theorem, we will evaluate the polynomial at [tex]\(x = -1\)[/tex].
The Factor Theorem states that for a polynomial [tex]\(P(x)\)[/tex], [tex]\(x - c\)[/tex] is a factor of [tex]\(P(x)\)[/tex] if and only if [tex]\(P(c) = 0\)[/tex]. Since we are given [tex]\(x + 1\)[/tex], we set [tex]\(c = -1\)[/tex] and calculate [tex]\(P(-1)\)[/tex]:
First, substitute [tex]\(x = -1\)[/tex] into [tex]\(P(x)\)[/tex]:
[tex]\[
P(-1) = -2(-1)^3 - 3(-1)^2 - 4(-1) + 6
\][/tex]
Now, calculate each term individually:
[tex]\[
(-1)^3 = -1 \quad \text{so} \quad -2(-1)^3 = -2(-1) = 2
\][/tex]
[tex]\[
(-1)^2 = 1 \quad \text{so} \quad -3(-1)^2 = -3(1) = -3
\][/tex]
[tex]\[
-4(-1) = 4
\][/tex]
Finally, we have the constant term:
[tex]\[
6
\][/tex]
Now, sum all these results:
[tex]\[
P(-1) = 2 - 3 + 4 + 6 = 9
\][/tex]
Since [tex]\(P(-1) = 9 \neq 0\)[/tex], we conclude:
[tex]\[
P(-1) = 9
\][/tex]
Hence, [tex]\(x + 1\)[/tex] is not a factor of [tex]\(P(x)\)[/tex].
To summarize:
[tex]\[
P(-1) = 9
\][/tex]
[tex]\(x+1\)[/tex] is not a factor of [tex]\(P(x)\)[/tex]
