Question

use synthetic division and the remainder theorem to find the indicated function value. f(x)=x^4 + 4x^3+7x^2 - 8x - 7; f(7) f(7)=

Explanation:

Step1: Set up synthetic division

Write the coefficients of the polynomial $f(x)=x^{4}+4x^{3}+7x^{2}-8x - 7$ which are $1,4,7,-8,-7$. And use $7$ for synthetic - division.

7	1	4	7	-8	-7
	1

Step2: Bring down the first coefficient

Bring down the first coefficient $1$.

7	1	4	7	-8	-7
	1

Step3: Multiply and add

Multiply $7\times1 = 7$ and add to the next coefficient: $4 + 7=11$.

7	1	4	7	-8	-7
	1	11

Step4: Repeat multiplication and addition

Multiply $7\times11 = 77$ and add to the next coefficient: $7+77 = 84$.

7	1	4	7	-8	-7
	1	11	84

Step5: Repeat again

Multiply $7\times84 = 588$ and add to the next coefficient: $-8 + 588=580$.

7	1	4	7	-8	-7
	1	11	84	580

Step6: One last multiplication and addition

Multiply $7\times580 = 4060$ and add to the last coefficient: $-7+4060 = 4053$.

7	1	4	7	-8	-7
	1	11	84	580	4053

By the Remainder Theorem, the remainder of the synthetic - division is the value of $f(7)$.

Answer:

$4053$