Use synthetic division and the Remainder Theorem to find the indicated function value.
[tex]f(x)=7 x^4+10 x^3+6 x^2-5 x+53 ; f(2)[/tex]
Answer (2)
Set up synthetic division with the coefficients of f ( x ) and c = 2 .
Perform synthetic division.
The remainder obtained from the synthetic division is the value of f ( 2 ) according to the Remainder Theorem.
The value of f ( 2 ) is 259 .
Explanation
Understanding the Problem We are given the polynomial f ( x ) = 7 x 4 + 10 x 3 + 6 x 2 − 5 x + 53 and we want to find f ( 2 ) using synthetic division and the Remainder Theorem. The Remainder Theorem tells us that if we divide the polynomial f ( x ) by x − c , the remainder will be f ( c ) . In this case, we want to find f ( 2 ) , so c = 2 . We will use synthetic division to divide f ( x ) by x − 2 .
Performing Synthetic Division We set up the synthetic division as follows:
2 | 7 10 6 -5 53
|________________
Now we perform the synthetic division:
Bring down the first coefficient (7):
2 | 7 10 6 -5 53
|________________
7
Multiply the value (2) by the result (7) and write it under the next coefficient (10):
2 | 7 10 6 -5 53
| 14
|________________
7
Add the numbers in the second column (10 + 14 = 24):
2 | 7 10 6 -5 53
| 14
|________________
7 24
Multiply the value (2) by the result (24) and write it under the next coefficient (6):
2 | 7 10 6 -5 53
| 14 48
|________________
7 24
Add the numbers in the third column (6 + 48 = 54):
2 | 7 10 6 -5 53
| 14 48
|________________
7 24 54
Multiply the value (2) by the result (54) and write it under the next coefficient (-5):
2 | 7 10 6 -5 53
| 14 48 108
|________________
7 24 54
Add the numbers in the fourth column (-5 + 108 = 103):
2 | 7 10 6 -5 53
| 14 48 108
|________________
7 24 54 103
Multiply the value (2) by the result (103) and write it under the next coefficient (53):
2 | 7 10 6 -5 53
| 14 48 108 206
|________________
7 24 54 103
Add the numbers in the last column (53 + 206 = 259):
2 | 7 10 6 -5 53
| 14 48 108 206
|________________
7 24 54 103 259
Applying the Remainder Theorem The remainder is 259. According to the Remainder Theorem, f ( 2 ) = 259 .
Final Answer Therefore, f ( 2 ) = 259 .
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 these polynomials at specific points is essential for rendering and displaying these shapes accurately. In engineering, polynomials are used to model physical systems, and synthetic division can help quickly determine the system's response at a particular input value. For example, if you have a polynomial representing the stress on a bridge as a function of temperature, you can use synthetic division to find the stress at a specific temperature.
We used synthetic division to evaluate the polynomial f ( x ) = 7 x 4 + 10 x 3 + 6 x 2 − 5 x + 53 at x = 2 . The remainder from the synthetic division process gave us f ( 2 ) = 259 . Thus, the value of the function at x = 2 is 259.
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