To determine the value of [tex]\(a\)[/tex] for which the polynomial [tex]\(4x^4 + 2x^3 - 3x^2 - ax - 28\)[/tex] has [tex]\(-2\)[/tex] as a zero, we need to substitute [tex]\(-2\)[/tex] for [tex]\(x\)[/tex] in the polynomial and set the equation equal to zero.
Let's substitute [tex]\(x = -2\)[/tex]:
[tex]\[4(-2)^4 + 2(-2)^3 - 3(-2)^2 - a(-2) - 28 = 0\][/tex]
Step-by-step calculation:
1. Calculate [tex]\(4(-2)^4\)[/tex]:
[tex]\[
4 \times (-2)^4 = 4 \times 16 = 64
\][/tex]
2. Calculate [tex]\(2(-2)^3\)[/tex]:
[tex]\[
2 \times (-2)^3 = 2 \times (-8) = -16
\][/tex]
3. Calculate [tex]\(-3(-2)^2\)[/tex]:
[tex]\[
-3 \times (-2)^2 = -3 \times 4 = -12
\][/tex]
4. The constant term is:
[tex]\[
-28
\][/tex]
5. Combine the calculated terms:
[tex]\[
64 - 16 - 12 - (-2a) - 28
\][/tex]
Combine and simplify the expression:
[tex]\[
64 - 16 - 12 - 28 + 2a = 0
\][/tex]
6. Sum the constants:
[tex]\[
64 - 16 - 12 - 28 = 8
\][/tex]
7. Incorporate the remaining terms to solve for [tex]\(a\)[/tex]:
[tex]\[
8 + 2a = 0
\][/tex]
8. Isolate [tex]\(a\)[/tex] by subtracting 8 from both sides:
[tex]\[
2a = -8
\][/tex]
9. Finally, solve for [tex]\(a\)[/tex] by dividing both sides by 2:
[tex]\[
a = -4
\][/tex]
Thus, the value of [tex]\(a\)[/tex] is [tex]\(-4\)[/tex].
So the correct answer is:
c) [tex]\(-4\)[/tex]
