To verify that [tex]\( -2 \)[/tex] is a zero of the polynomial [tex]\( h(x) = x^3 - 4x^2 - 8x + 8 \)[/tex], we need to substitute [tex]\( x = -2 \)[/tex] into the polynomial and check if the result equals zero.
Here is the step-by-step process:
1. Substitute [tex]\( x = -2 \)[/tex] into the polynomial:
[tex]\[
h(-2) = (-2)^3 - 4(-2)^2 - 8(-2) + 8
\][/tex]
2. Calculate each term separately:
- The first term is [tex]\( (-2)^3 \)[/tex]:
[tex]\[
(-2)^3 = -8
\][/tex]
- The second term is [tex]\( -4(-2)^2 \)[/tex]:
[tex]\[
(-2)^2 = 4 \quad \text{and thus} \quad -4 \times 4 = -16
\][/tex]
- The third term is [tex]\( -8(-2) \)[/tex]:
[tex]\[
-8 \times (-2) = 16
\][/tex]
- The fourth term is [tex]\( +8 \)[/tex]:
[tex]\[
+8
\][/tex]
3. Combine the results of each term:
[tex]\[
h(-2) = -8 - 16 + 16 + 8
\][/tex]
4. Simplify the expression:
- Adding the first two terms:
[tex]\[
-8 - 16 = -24
\][/tex]
- Adding the next term:
[tex]\[
-24 + 16 = -8
\][/tex]
- Finally, adding the last term:
[tex]\[
-8 + 8 = 0
\][/tex]
Since [tex]\( h(-2) = 0 \)[/tex], we have verified that [tex]\( -2 \)[/tex] is indeed a zero of the polynomial [tex]\( h(x) = x^3 - 4x^2 - 8x + 8 \)[/tex].
