Certainly! Let's use the factor theorem to prove that [tex]\((x - 2)\)[/tex] is a factor of the polynomial [tex]\(2x^3 - 7x - 2\)[/tex].
### Step-by-Step Solution:
1. Factor Theorem:
The factor theorem states that [tex]\((x - c)\)[/tex] is a factor of a polynomial [tex]\(f(x)\)[/tex] if and only if [tex]\(f(c) = 0\)[/tex]. To apply this theorem, we need to substitute [tex]\(x = 2\)[/tex] into the polynomial and check if the result is zero.
2. Substitute [tex]\(x = 2\)[/tex] into the Polynomial:
Let's find the value of the polynomial [tex]\(2x^3 - 7x - 2\)[/tex] at [tex]\(x = 2\)[/tex].
[tex]\[
f(x) = 2x^3 - 7x - 2
\][/tex]
Substituting [tex]\(x = 2\)[/tex],
[tex]\[
f(2) = 2(2)^3 - 7(2) - 2
\][/tex]
3. Calculate [tex]\(f(2)\)[/tex]:
Now, compute the values step-by-step:
[tex]\[
2(2)^3 = 2 \times 8 = 16
\][/tex]
[tex]\[
7(2) = 14
\][/tex]
[tex]\[
f(2) = 16 - 14 - 2
\][/tex]
[tex]\[
f(2) = 16 - 14 - 2 = 0
\][/tex]
4. Determine if [tex]\((x - 2)\)[/tex] is a Factor:
Since [tex]\(f(2) = 0\)[/tex] evaluates to zero, by the factor theorem, [tex]\((x - 2)\)[/tex] is indeed a factor of the polynomial [tex]\(2x^3 - 7x - 2\)[/tex].
### Conclusion:
We have shown that [tex]\(f(2) = 0\)[/tex]. Therefore, by the factor theorem, [tex]\((x - 2)\)[/tex] is a factor of the polynomial [tex]\(2x^3 - 7x - 2\)[/tex].
