Question

Given that f(x)=9x^2−50, apply the Remainder Theorem to find the remainder of (9x^2−50)÷(x−7/3).

Answer 1

To apply the Remainder Theorem, we will evaluate the function \( f(x) = 9x^2 - 50 \) at \( x = \frac{7}{3} \). The Remainder Theorem states that the remainder of the polynomial \( f(x) \) when divided by \( x - c \) is equal to \( f(c) \).

1. **Finding \( f\left(\frac{7}{3}\right) \)**:
\[
f\left(\frac{7}{3}\right) = 9\left(\frac{7}{3}\right)^2 - 50
\]

2. **Calculating \( \left(\frac{7}{3}\right)^2 \)**:
\[
\left(\frac{7}{3}\right)^2 = \frac{49}{9}
\]

3. **Substituting into \( f\left(\frac{7}{3}\right) \)**:
\[
f\left(\frac{7}{3}\right) = 9 \cdot \frac{49}{9} - 50
\]
The \( 9 \) and the \( \frac{9}{1} \) cancel out:
\[
= 49 - 50 = -1
\]

Thus, the remainder of \( (9x^2 - 50) \) divided by \( \left(x - \frac{7}{3}\right) \) is \(\boxed{-1}\).