Use synthetic division and the Remainder Theorem to evaluate. [tex]f(x)=x^5+2 x^4+5 x^3+3 ; f(-4)[/tex]
Answer (2)
To evaluate the polynomial f ( x ) = x 5 + 2 x 4 + 5 x 3 + 3 at x = − 4 using synthetic division, we find the remainder is -829. According to the Remainder Theorem, this means f ( − 4 ) = − 829 . Therefore, the answer is − 829 .
;
Use synthetic division to divide the polynomial f ( x ) = x 5 + 2 x 4 + 5 x 3 + 3 by x + 4 .
Perform the synthetic division to find the remainder.
Apply the Remainder Theorem, which states that f ( − 4 ) is equal to the remainder.
The remainder is -829, so f ( − 4 ) = − 829 .
Explanation
Understanding the Problem We are given the polynomial f ( x ) = x 5 + 2 x 4 + 5 x 3 + 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 .
Performing Synthetic Division We will use synthetic division to divide f ( x ) by x + 4 . The coefficients of f ( x ) are 1, 2, 5, 0, 0, and 3. We set up the synthetic division as follows:
-4 | 1 2 5 0 0 3
| -4 8 -52 208 -832
--------------------------
1 -2 13 -52 208 -829
We bring down the first coefficient (1). Then, we multiply -4 by 1 to get -4, and add it to 2 to get -2. Next, we multiply -4 by -2 to get 8, and add it to 5 to get 13. We continue this process until we reach the last coefficient.
Applying the Remainder Theorem The last number in the bottom row is the remainder, which is -829. According to the Remainder Theorem, f ( − 4 ) = − 829 .
Final Answer Therefore, f ( − 4 ) = − 829 .
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 at specific points. For example, in engineering, you might use these techniques to analyze the behavior of a system modeled by a polynomial function, such as determining the stability of a control system or predicting the response of a circuit to different inputs. By efficiently evaluating the polynomial at specific values, you can gain insights into the system's characteristics and make informed decisions about its design and operation.
