Answer :
To determine the value of [tex]\( h(-6) \)[/tex] given that [tex]\( x+6 \)[/tex] is a factor of the polynomial function [tex]\( h \)[/tex], we need to apply the Factor Theorem.
### Step-by-Step Solution:
1. Understand the Factor Theorem: The Factor Theorem states that for a polynomial [tex]\( h(x) \)[/tex], if [tex]\( (x - c) \)[/tex] is a factor of [tex]\( h(x) \)[/tex], then [tex]\( h(c) = 0 \)[/tex]. Here, instead of [tex]\( (x - c) \)[/tex], we are given [tex]\( (x + 6) \)[/tex] as a factor of [tex]\( h(x) \)[/tex].
2. Rewrite the Factor: To find [tex]\( c \)[/tex], we need to rewrite [tex]\( (x + 6) \)[/tex] in the form [tex]\( (x - c) \)[/tex]. Notice that:
[tex]\[ (x + 6) = (x - (-6)) \][/tex]
Therefore, [tex]\( c = -6 \)[/tex].
3. Apply the Factor Theorem: Since [tex]\( (x + 6) \)[/tex] is a factor of [tex]\( h(x) \)[/tex], this means that:
[tex]\[ h(-6) = 0 \][/tex]
Therefore, the value of [tex]\( h(-6) \)[/tex] is [tex]\( \boxed{0} \)[/tex].
### Step-by-Step Solution:
1. Understand the Factor Theorem: The Factor Theorem states that for a polynomial [tex]\( h(x) \)[/tex], if [tex]\( (x - c) \)[/tex] is a factor of [tex]\( h(x) \)[/tex], then [tex]\( h(c) = 0 \)[/tex]. Here, instead of [tex]\( (x - c) \)[/tex], we are given [tex]\( (x + 6) \)[/tex] as a factor of [tex]\( h(x) \)[/tex].
2. Rewrite the Factor: To find [tex]\( c \)[/tex], we need to rewrite [tex]\( (x + 6) \)[/tex] in the form [tex]\( (x - c) \)[/tex]. Notice that:
[tex]\[ (x + 6) = (x - (-6)) \][/tex]
Therefore, [tex]\( c = -6 \)[/tex].
3. Apply the Factor Theorem: Since [tex]\( (x + 6) \)[/tex] is a factor of [tex]\( h(x) \)[/tex], this means that:
[tex]\[ h(-6) = 0 \][/tex]
Therefore, the value of [tex]\( h(-6) \)[/tex] is [tex]\( \boxed{0} \)[/tex].
