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Question 2 (Essay Worth 4 points)
(H1.01 MC)

What is the remainder when [tex]$f(x) = x^3 + 5x^2 - 32x - 7$[/tex] is divided by [tex]$x - 4$[/tex]? Show your work.


Sagot :

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]