To find the remainder when the polynomial [tex]\( f(x) = x^3 + 5x^2 - 32x - 7 \)[/tex] is divided by [tex]\( x - 4 \)[/tex], we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial [tex]\( f(x) \)[/tex] by a linear divisor [tex]\( x - c \)[/tex] is [tex]\( f(c) \)[/tex].
In this case, our polynomial is [tex]\( f(x) \)[/tex] and our divisor is [tex]\( x - 4 \)[/tex]. Therefore, we need to find [tex]\( f(4) \)[/tex].
Let's evaluate [tex]\( f(4) \)[/tex]:
1. Start with the given polynomial:
[tex]\[
f(x) = x^3 + 5x^2 - 32x - 7
\][/tex]
2. Substitute [tex]\( x = 4 \)[/tex] into the polynomial:
[tex]\[
f(4) = (4)^3 + 5(4)^2 - 32(4) - 7
\][/tex]
3. Calculate each term:
- [tex]\( (4)^3 = 64 \)[/tex]
- [tex]\( 5(4)^2 = 5 \times 16 = 80 \)[/tex]
- [tex]\( 32(4) = 128 \)[/tex]
4. Substitute these values back into the expression:
[tex]\[
f(4) = 64 + 80 - 128 - 7
\][/tex]
5. Simplify the expression step by step:
- [tex]\( 64 + 80 = 144 \)[/tex]
- [tex]\( 144 - 128 = 16 \)[/tex]
- [tex]\( 16 - 7 = 9 \)[/tex]
Thus, the remainder when [tex]\( f(x) = x^3 + 5x^2 - 32x - 7 \)[/tex] is divided by [tex]\( x - 4 \)[/tex] is
[tex]\[
\boxed{9}
\][/tex]
