Question
Given that f(x)=16x^2-20
, apply the Remainder Theorem to find the remainder of (16x^2-20) divided (x-3/4)
.
A
124124124
B
111111
C
−11-11−11
D
−8-8−8
Answers
GPT-4o mini
To find the remainder of the polynomial \( f(x) = 16x^2 - 20 \) when divided by \( x - \frac{3}{4} \), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).
In this case, \( c = \frac{3}{4} \).
Now, we will evaluate the polynomial \( f(x) \) at \( x = \frac{3}{4} \):
\[
f\left(\frac{3}{4}\right) = 16\left(\frac{3}{4}\right)^2 - 20
\]
Calculating \( \left(\frac{3}{4}\right)^2 \):
\[
\left(\frac{3}{4}\right)^2 = \frac{9}{16}
\]
Now substitute this back into the polynomial:
\[
f\left(\frac{3}{4}\right) = 16 \cdot \frac{9}{16} - 20
\]
\[
= 9 - 20
\]
\[
= -11
\]
Therefore, the remainder when \( 16x^2 - 20 \) is divided by \( x - \frac{3}{4} \) is \( -11 \).
Thus, the correct answer is:
C \(-11\)
In this case, \( c = \frac{3}{4} \).
Now, we will evaluate the polynomial \( f(x) \) at \( x = \frac{3}{4} \):
\[
f\left(\frac{3}{4}\right) = 16\left(\frac{3}{4}\right)^2 - 20
\]
Calculating \( \left(\frac{3}{4}\right)^2 \):
\[
\left(\frac{3}{4}\right)^2 = \frac{9}{16}
\]
Now substitute this back into the polynomial:
\[
f\left(\frac{3}{4}\right) = 16 \cdot \frac{9}{16} - 20
\]
\[
= 9 - 20
\]
\[
= -11
\]
Therefore, the remainder when \( 16x^2 - 20 \) is divided by \( x - \frac{3}{4} \) is \( -11 \).
Thus, the correct answer is:
C \(-11\)