To determine whether [tex]\( x-3 \)[/tex] is a factor of the polynomial [tex]\( P(x) = 2x^3 - 4x^2 - 4x + 5 \)[/tex] using the Factor Theorem, follow these steps:
1. Understand the Factor Theorem: The Factor Theorem states that [tex]\( x - a \)[/tex] is a factor of a polynomial [tex]\( P(x) \)[/tex] if and only if [tex]\( P(a) = 0 \)[/tex].
2. Identify the Value to Evaluate: Here, we need to check if [tex]\( x - 3 \)[/tex] is a factor. According to the Factor Theorem, this requires us to evaluate the polynomial [tex]\( P(x) \)[/tex] at [tex]\( x = 3 \)[/tex].
3. Evaluate [tex]\( P(3) \)[/tex]:
[tex]\[
P(3) = 2(3)^3 - 4(3)^2 - 4(3) + 5
\][/tex]
4. Compute Each Term:
- Calculate [tex]\( 3^3 \)[/tex]:
[tex]\[
3^3 = 27
\][/tex]
- Calculate [tex]\( 2 \times 27 \)[/tex]:
[tex]\[
2 \times 27 = 54
\][/tex]
- Calculate [tex]\( 3^2 \)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
- Calculate [tex]\( 4 \times 9 \)[/tex]:
[tex]\[
4 \times 9 = 36
\][/tex]
- Calculate [tex]\( 4 \times 3 \)[/tex]:
[tex]\[
4 \times 3 = 12
\][/tex]
5. Substitute and Combine the Terms:
[tex]\[
P(3) = 54 - 36 - 12 + 5
\][/tex]
6. Perform the Arithmetic:
[tex]\[
54 - 36 = 18
\][/tex]
[tex]\[
18 - 12 = 6
\][/tex]
[tex]\[
6 + 5 = 11
\][/tex]
Thus, [tex]\( P(3) = 11 \)[/tex].
7. Apply the Factor Theorem:
Since [tex]\( P(3) = 11 \neq 0 \)[/tex], according to the Factor Theorem, [tex]\( x - 3 \)[/tex] is not a factor of [tex]\( P(x) \)[/tex].
Hence,
[tex]\[ P(3) = 11 \][/tex]
[tex]\[ x-3 \text{ is not a factor of } P(x) \][/tex]
