Use synthetic division and the Remainder Theorem to find the indicated value. [tex]f(x)=4 x^3-7 x^2-5 x+24 ; f(-2)[/tex]
Answer (2)
Using synthetic division, we found that f ( − 2 ) for the polynomial f ( x ) = 4 x 3 − 7 x 2 − 5 x + 24 is − 26 by applying the Remainder Theorem. The remainder obtained through the division process confirms this value.
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We are given the polynomial f ( x ) = 4 x 3 − 7 x 2 − 5 x + 24 and asked to find f ( − 2 ) using synthetic division and the Remainder Theorem.
Perform synthetic division to divide f ( x ) by x + 2 .
The remainder obtained from the synthetic division is -26.
By the Remainder Theorem, f ( − 2 ) = − 26 , so the final answer is − 26 .
Explanation
Understanding the Problem We are given the polynomial f ( x ) = 4 x 3 − 7 x 2 − 5 x + 24 and we want to find the value of f ( − 2 ) using synthetic division and the Remainder Theorem. The Remainder Theorem states that if we divide a polynomial f ( x ) by x − c , then the remainder is f ( c ) . In this case, we want to find f ( − 2 ) , so c = − 2 .
Performing Synthetic Division We will use synthetic division to divide f ( x ) by x − ( − 2 ) = x + 2 . The coefficients of f ( x ) are 4, -7, -5, and 24. We set up the synthetic division as follows:
-2 | 4 -7 -5 24
| -8 30 -50
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4 -15 25 -26
We bring down the first coefficient, 4. Then we multiply -2 by 4 to get -8, and add it to -7 to get -15. Next, we multiply -2 by -15 to get 30, and add it to -5 to get 25. Finally, we multiply -2 by 25 to get -50, and add it to 24 to get -26.
Applying the Remainder Theorem The remainder from the synthetic division is -26. According to the Remainder Theorem, this is the value of f ( − 2 ) . Therefore, f ( − 2 ) = − 26 .
Final Answer Thus, f ( − 2 ) = − 26 .
Examples
Synthetic division and the Remainder Theorem are useful in various applications, such as finding the roots of polynomials, simplifying algebraic expressions, and solving engineering problems. For example, in control systems, engineers use polynomials to model the behavior of a system. By finding the roots of the polynomial, they can determine the stability of the system. Also, in computer graphics, polynomials are used to represent curves and surfaces. Synthetic division can be used to efficiently evaluate these polynomials at different points, which is useful for rendering the graphics.
