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 of (f(x)). We use polynomial long - division or synthetic division to divide (x^{3}-4x^{2}-20x + 48) by ((x - 6)). Using synthetic division: The coefficients of (f(x)) are (1,-4,-20,48). Bring down the first coefficient (1): Multiply (6\times1 = 6), add to the second coefficient: (-4+6 = 2) Multiply (6\times2=12), add to the third coefficient: (-20 + 12=-8) Multiply (6\times(-8)=-48), add to the fourth coefficient: (48-48 = 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))