To determine if [tex]\( x-3 \)[/tex] is a factor of the polynomial [tex]\( P(x) = x^3 - 7x^2 + 15x - 9 \)[/tex], we can use the Factor Theorem. According to the Factor Theorem, [tex]\( x-a \)[/tex] is a factor of [tex]\( P(x) \)[/tex] if and only if [tex]\( P(a) = 0 \)[/tex].
Here’s the step-by-step process to verify this:
1. Identify the value of [tex]\( a \)[/tex]:
Since we are checking if [tex]\( x-3 \)[/tex] is a factor, [tex]\( a \)[/tex] is 3.
2. Substitute [tex]\( a = 3 \)[/tex] into the polynomial [tex]\( P(x) \)[/tex]:
We need to calculate [tex]\( P(3) \)[/tex].
[tex]\[
P(3) = (3)^3 - 7(3)^2 + 15(3) - 9
\][/tex]
3. Compute [tex]\( P(3) \)[/tex]:
- [tex]\( 3^3 = 27 \)[/tex]
- [tex]\( 7(3^2) = 7(9) = 63 \)[/tex]
- [tex]\( 15(3) = 45 \)[/tex]
- Combining these, we get:
[tex]\[
P(3) = 27 - 63 + 45 - 9
\][/tex]
4. Simplify the expression:
[tex]\[
P(3) = 27 - 63 + 45 - 9 = (27 + 45 - 63 - 9)
\][/tex]
[tex]\[
= 72 - 72
\][/tex]
[tex]\[
= 0
\][/tex]
5. Conclusion:
Since [tex]\( P(3) = 0 \)[/tex], according to the Factor Theorem, [tex]\( x-3 \)[/tex] is indeed a factor of the polynomial [tex]\( P(x) = x^3 - 7x^2 + 15x - 9 \)[/tex].
Therefore, the answer is:
A. True
