Question

use synthetic division and the remainder theorem to find the indicated function value. f(x)=x^4 + 3x^3 + 4x^2 - 6x - 3; f(5) f(6) =

Explanation:

Step1: Set up synthetic division

Write the coefficients of the polynomial $f(x)=x^{4}+3x^{3}+4x^{2}-6x - 3$ which are $1,3,4,-6,-3$. And the value $c = 5$ for synthetic - division.

5	1	3	4	-6	-3

Step2: Bring down the first coefficient

Bring down the first coefficient 1.

5	1	3	4	-6	-3
	1

Step3: Multiply and add

Multiply 5 by 1 to get 5, add to 3: $3 + 5=8$.

5	1	3	4	-6	-3
	1	8

Multiply 5 by 8 to get 40, add to 4: $4+40 = 44$.

5	1	3	4	-6	-3
	1	8	44

Multiply 5 by 44 to get 220, add to - 6: $-6+220 = 214$.

5	1	3	4	-6	-3
	1	8	44	214

Multiply 5 by 214 to get 1070, add to - 3: $-3 + 1070=1067$.

5	1	3	4	-6	-3
	1	8	44	214	1067

Step4: Apply the Remainder Theorem

By the Remainder Theorem, when we divide a polynomial $f(x)$ by $x - c$, the remainder is $f(c)$. Here, the remainder of the synthetic - division is 1067, so $f(5)=1067$.

Answer:

1067