Use synthetic division and the Remainder Theorem to find the value. [tex]f(x)=9 x^4+11 x^3+6 x^2-6 x+23 ; f(-3)[/tex]

Using synthetic division on the polynomial f ( x ) = 9 x 4 + 11 x 3 + 6 x 2 − 6 x + 23 with the value x = − 3 , we find that the remainder is 527. Thus, according to the Remainder Theorem, f ( − 3 ) = 527 .
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Answered by Anonymous | 2025-07-04

Set up synthetic division with -3 and the coefficients of the polynomial.
Perform synthetic division to find the remainder.
Apply the Remainder Theorem to determine that f ( − 3 ) equals the remainder.
Conclude that f ( − 3 ) = 527 ​ .

Explanation

Understanding the Problem We are asked to find the value of the polynomial f ( x ) = 9 x 4 + 11 x 3 + 6 x 2 − 6 x + 23 at x = − 3 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 = − 3 .

Setting up Synthetic Division We will use synthetic division to divide the polynomial by x − ( − 3 ) = x + 3 . We set up the synthetic division as follows:
| 9 11 6 -6 23


-3 |________________________
|

Performing Synthetic Division Now we perform the synthetic division:

Bring down the first coefficient (9).

Multiply -3 by 9 and write the result (-27) under 11.

Add 11 and -27 to get -16.

Multiply -3 by -16 and write the result (48) under 6.

Add 6 and 48 to get 54.

Multiply -3 by 54 and write the result (-162) under -6.

Add -6 and -162 to get -168.

Multiply -3 by -168 and write the result (504) under 23.

Add 23 and 504 to get 527.


The synthetic division looks like this:
| 9 11 6 -6 23 -3 | -27 48 -162 504 |________________________
| 9 -16 54 -168 527

Applying the Remainder Theorem The remainder is 527. According to the Remainder Theorem, f ( − 3 ) = 527 .

Final Answer Therefore, f ( − 3 ) = 527 .


Examples
Synthetic division and the Remainder Theorem are useful in various real-world applications, such as computer graphics, where polynomials are used to model curves and surfaces. Evaluating polynomials quickly is essential for rendering images and animations efficiently. In engineering, synthetic division can be used to analyze the stability of systems modeled by polynomials. For example, determining if a system's response remains bounded over time involves finding the roots of a characteristic polynomial, which can be simplified using synthetic division.

Answered by GinnyAnswer | 2025-07-04