Question

Based on the polynomial remainder theorem, what is the value of the function when x = 3?

f(x)=x4+3x3−6x2−12x−8

Answer 1

64

Explanation:

Evaluate x^4 + 3 x^3 - 6 x^2 - 12 x - 8 where x = 3:

x^4 + 3 x^3 - 6 x^2 - 12 x - 8 = 3^4 + 3×3^3 - 6×3^2 - 12×3 - 8

3^3 = 3×3^2:

3^4 + 3×3×3^2 - 6×3^2 - 12×3 - 8

3^2 = 9:

3^4 + 3×3×9 - 6×3^2 - 12×3 - 8

3×9 = 27:

3^4 + 3×27 - 6×3^2 - 12×3 - 8

3^2 = 9:

3^4 + 3×27 - 69 - 12×3 - 8

3^4 = (3^2)^2:

(3^2)^2 + 3×27 - 6×9 - 12×3 - 8

3^2 = 9:

9^2 + 3×27 - 6×9 - 12×3 - 8

9^2 = 81:

81 + 3×27 - 6×9 - 12×3 - 8

3×27 = 81:

81 + 81 - 6×9 - 12×3 - 8

-6×9 = -54:

81 + 81 + -54 - 12×3 - 8

-12×3 = -36:

81 + 81 - 54 + -36 - 8

81 + 81 - 54 - 36 - 8 = (81 + 81) - (54 + 36 + 8):

(81 + 81) - (54 + 36 + 8)

| 8 | 1

+ | 8 | 1

1 | 6 | 2:

162 - (54 + 36 + 8)

| 1 |

| 5 | 4

| 3 | 6

+ | | 8

| 9 | 8:

162 - 98

| | 15 |

| 0 | 5 | 12

| 1 | 6 | 2

- | | 9 | 8

| 0 | 6 | 4:

Answer: 64

Answer 2

Answer: 64

Explanation:

f(x) = x⁴ + 3x³ - 6x² -12x -8

when x= 3

f(3) = (3)⁴ + 3(3)³ - 6(3)² -12(3) -8

=81 + 3(27) - 6(9)- 12(3) - 8

= 81 + 81 - 54 - 36 - 8

= 64

Therefore, when x= 3 the function becomes; f(3)=64