Use synthetic division and the Remainder Theorem to find the value.
[tex]f(x)=x^4-8 x^3+2 x^2+7 x-3 ; f(-4)[/tex]
Answer (2)
To find f ( − 4 ) for the polynomial f ( x ) = x 4 − 8 x 3 + 2 x 2 + 7 x − 3 , we use synthetic division with c = − 4 . After performing the division, we find that the remainder is 769 , hence f ( − 4 ) = 769 .
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Set up synthetic division with the coefficients of f ( x ) and c = − 4 .
Perform synthetic division to find the remainder.
Apply the Remainder Theorem to determine f ( − 4 ) .
The value of f ( − 4 ) is 769 .
Explanation
Understanding the Problem We are given the polynomial f ( x ) = x 4 − 8 x 3 + 2 x 2 + 7 x − 3 and we want to find the value of f ( − 4 ) using synthetic division and the Remainder Theorem. The Remainder Theorem states that if we divide a polynomial f ( x ) by x − c , the remainder is f ( c ) . In this case, c = − 4 .
Setting up Synthetic Division We set up the synthetic division with c = − 4 and the coefficients of f ( x ) , which are 1, -8, 2, 7, and -3.
Performing Synthetic Division Now, we perform the synthetic division:
-4 | 1 -8 2 7 -3 | -4 48 -200 769 |------------------------ 1 -12 50 -193 766
The remainder is 769.
Applying the Remainder Theorem According to the Remainder Theorem, the remainder obtained from the synthetic division is the value of f ( − 4 ) . Therefore, f ( − 4 ) = 769 .
Final Answer Thus, the value of f ( − 4 ) is 769.
Examples
Synthetic division and the Remainder Theorem are useful in various applications, such as finding roots of polynomials, simplifying complex expressions, and evaluating polynomial functions efficiently. For instance, in engineering, when analyzing the behavior of a system modeled by a polynomial, evaluating the polynomial at specific points is crucial. Synthetic division provides a quick way to determine the output of the system for a given input, aiding in system design and optimization. Imagine designing a bridge where the load distribution is modeled by a polynomial function; using synthetic division, engineers can quickly assess the stress at different points along the bridge's structure.
