Questions: Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=3x^3-8x^2+3x-4; f(2) f(2)=

Solution Steps

We start with the polynomial \( f(x) = 3x^{3} - 8x^{2} + 3x - 4 \) and we want to evaluate \( f(2) \) using synthetic division. We divide \( f(x) \) by \( x - 2 \).

1. Divide \( 3x^{3} \) by \( x \), resulting in \( 3x^{2} \). The remaining polynomial is \( -2x^{2} + 3x - 4 \).
2. Divide \( -2x^{2} \) by \( x \), resulting in \( -2x \). The remaining polynomial is \( -x - 4 \).
3. Divide \( -x \) by \( x \), resulting in \( -1 \). The remaining polynomial is \( -6 \).

Thus, the quotient is \( 3x^{2} - 2x - 1 \) and the remainder is \( -6 \).

We can express the division as: \[ \frac{3x^{3} - 8x^{2} + 3x - 4}{x - 2} = 3x^{2} - 2x - 1 - \frac{6}{x - 2} \]

According to the Remainder Theorem, the value of \( f(2) \) is equal to the remainder when \( f(x) \) is divided by \( x - 2 \). Therefore, we have: \[ f(2) = -6 \]

Final Answer