one root of (f(x)=x^{3}-4x^{2}-20x + 48) is (x = 6). what are all the factors of the function? use the…

one root of (f(x)=x^{3}-4x^{2}-20x + 48) is (x = 6). what are all the factors of the function? use the remainder theorem.\n((x + 6)(x + 8)\n(x - 6)(x - 8)\n(x - 2)(x + 4)(x - 6)\n(x + 2)(x - 4)(x + 6))

one root of (f(x)=x^{3}-4x^{2}-20x + 48) is (x = 6). what are all the factors of the function? use the remainder theorem.\n((x + 6)(x + 8)\n(x - 6)(x - 8)\n(x - 2)(x + 4)(x - 6)\n(x + 2)(x - 4)(x + 6))

Answer

Explanation:

Step1: Apply Remainder Theorem

Since (x = 6) is a root of (f(x)=x^{3}-4x^{2}-20x + 48), then ((x - 6)) is a factor. Use polynomial long - division or synthetic division to divide (x^{3}-4x^{2}-20x + 48) by ((x - 6)). Using synthetic division: |6| 1 -4 -20 48| | | 6 12 -48| | | 1 2 -8 0| The quotient is (x^{2}+2x - 8).

Step2: Factor the quotient

Factor (x^{2}+2x - 8). We need to find two numbers that multiply to (-8) and add up to (2). The numbers are (4) and (-2). So (x^{2}+2x - 8=(x + 4)(x - 2)).

Answer:

C. ((x - 2)(x + 4)(x - 6))