Answer :
Sure! Let's tackle this problem step-by-step.
We are given the polynomial function:
[tex]\[ f(x) = x^4 + 4x^3 - 7x^2 - 22x + 24 \][/tex]
and it's mentioned that [tex]\( x + 3 \)[/tex] is a factor of [tex]\( f(x) \)[/tex]. If [tex]\( x + 3 \)[/tex] is a factor, then setting [tex]\( x = -3 \)[/tex] should make [tex]\( f(x) \)[/tex] equal to zero. This is due to the Factor Theorem, which states that if [tex]\( x - c \)[/tex] is a factor of a polynomial, then [tex]\( f(c) = 0 \)[/tex].
To find the value of [tex]\( f(-3) \)[/tex], we simply substitute [tex]\( -3 \)[/tex] into the function:
1. Calculate [tex]\( (-3)^4 \)[/tex]:
[tex]\[ (-3)^4 = 81 \][/tex]
2. Calculate [tex]\( 4(-3)^3 \)[/tex]:
[tex]\[ 4 \times (-27) = -108 \][/tex]
3. Calculate [tex]\( -7(-3)^2 \)[/tex]:
[tex]\[ -7 \times 9 = -63 \][/tex]
4. Calculate [tex]\( -22(-3) \)[/tex]:
[tex]\[ 66 \][/tex]
5. Add all the calculated terms along with the constant:
[tex]\[ 81 - 108 - 63 + 66 + 24 \][/tex]
6. Simplify:
[tex]\[ = (81 + 66 + 24) - (108 + 63) \][/tex]
[tex]\[ = 171 - 171 \][/tex]
[tex]\[ = 0 \][/tex]
So, the value of [tex]\( f(-3) \)[/tex] is [tex]\( 0 \)[/tex].
Therefore, the correct answer is [tex]\( \boxed{0} \)[/tex].
We are given the polynomial function:
[tex]\[ f(x) = x^4 + 4x^3 - 7x^2 - 22x + 24 \][/tex]
and it's mentioned that [tex]\( x + 3 \)[/tex] is a factor of [tex]\( f(x) \)[/tex]. If [tex]\( x + 3 \)[/tex] is a factor, then setting [tex]\( x = -3 \)[/tex] should make [tex]\( f(x) \)[/tex] equal to zero. This is due to the Factor Theorem, which states that if [tex]\( x - c \)[/tex] is a factor of a polynomial, then [tex]\( f(c) = 0 \)[/tex].
To find the value of [tex]\( f(-3) \)[/tex], we simply substitute [tex]\( -3 \)[/tex] into the function:
1. Calculate [tex]\( (-3)^4 \)[/tex]:
[tex]\[ (-3)^4 = 81 \][/tex]
2. Calculate [tex]\( 4(-3)^3 \)[/tex]:
[tex]\[ 4 \times (-27) = -108 \][/tex]
3. Calculate [tex]\( -7(-3)^2 \)[/tex]:
[tex]\[ -7 \times 9 = -63 \][/tex]
4. Calculate [tex]\( -22(-3) \)[/tex]:
[tex]\[ 66 \][/tex]
5. Add all the calculated terms along with the constant:
[tex]\[ 81 - 108 - 63 + 66 + 24 \][/tex]
6. Simplify:
[tex]\[ = (81 + 66 + 24) - (108 + 63) \][/tex]
[tex]\[ = 171 - 171 \][/tex]
[tex]\[ = 0 \][/tex]
So, the value of [tex]\( f(-3) \)[/tex] is [tex]\( 0 \)[/tex].
Therefore, the correct answer is [tex]\( \boxed{0} \)[/tex].
