Questions: Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x^4+3x^3+6x^2-7x-7; f(4) f(4)=

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x^4+3x^3+6x^2-7x-7; f(4) f(4)=
Transcript text: Use synthetic division and the Remainder Theorem to find the indicated function value. \[ f(x)=x^{4}+3 x^{3}+6 x^{2}-7 x-7 ; f(4) \] \[ f(4)= \]
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Solution

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Solution Steps

To find the value of \( f(4) \) using synthetic division and the Remainder Theorem, we will perform synthetic division of the polynomial \( f(x) = x^4 + 3x^3 + 6x^2 - 7x - 7 \) by \( x - 4 \). The remainder of this division will give us the value of \( f(4) \).

Step 1: Define the Polynomial

We are given the polynomial function \( f(x) = x^4 + 3x^3 + 6x^2 - 7x - 7 \).

Step 2: Apply Synthetic Division

To find \( f(4) \), we perform synthetic division of \( f(x) \) by \( x - 4 \). The coefficients of the polynomial are \( [1, 3, 6, -7, -7] \).

Step 3: Calculate the Remainder

After performing synthetic division, we find that the remainder is \( 509 \). According to the Remainder Theorem, this remainder represents the value of the polynomial at \( x = 4 \).

Final Answer

Thus, the value of \( f(4) \) is \( \boxed{509} \).

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