To determine whether [tex]\( x - 1 \)[/tex] is a factor of [tex]\( P(x) = -x^4 + x^3 + 6x^2 - 9 \)[/tex], we can use the Factor Theorem. The Factor Theorem states that [tex]\( x - c \)[/tex] is a factor of the polynomial [tex]\( P(x) \)[/tex] if and only if [tex]\( P(c) = 0 \)[/tex].
In this case, we need to evaluate [tex]\( P(1) \)[/tex] and check if it equals zero.
First, we evaluate the polynomial at [tex]\( x = 1 \)[/tex]:
[tex]\[
P(1) = - (1)^4 + (1)^3 + 6(1)^2 - 9
\][/tex]
Calculate each term step-by-step:
[tex]\[
(1)^4 = 1 \implies - (1)^4 = -1
\][/tex]
[tex]\[
(1)^3 = 1
\][/tex]
[tex]\[
6(1)^2 = 6
\][/tex]
[tex]\[
-9 \text{ (constant term)}
\][/tex]
Combine all the terms:
[tex]\[
P(1) = -1 + 1 + 6 - 9
\][/tex]
Simplify the expression:
[tex]\[
P(1) = 7 - 9 = -3
\][/tex]
So, [tex]\( P(1) = -3 \)[/tex].
Since [tex]\( P(1) \neq 0 \)[/tex], by the Factor Theorem, [tex]\( x - 1 \)[/tex] is not a factor of [tex]\( P(x) \)[/tex].
Thus, the detailed answer is:
[tex]\[
P(1) = -3
\][/tex]
And
[tex]\[
x-1 \text{ is not a factor of } P(x).
\][/tex]
