Given that x=2
is a zero of f(x)=x3−3x2−10x+24
, apply the Factor Theorem and factor f(x)
into linear factors.(1 point)
Responses
f(x)=(x−2)(x+3)(x−4)
f left parenthesis x right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x plus 3 right parenthesis left parenthesis x minus 4 right parenthesis
f(x)=(x−2)(x2 −x−12)
f left parenthesis x right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x squared minus x minus 12 right parenthesis
f(x)=(x+2)(x+3)(x−4)
f left parenthesis x right parenthesis equals left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis left parenthesis x minus 4 right parenthesis
f(x)=(x−2)(x−3)(x−4)
f(x)=(x−2)(x−3)(x−4)
f(x)=(x−2)(x−3)(x−4)
To apply the Factor Theorem and factor the given function f(x) = x^3 - 3x^2 - 10x + 24, we need to use the zero x = 2.
The Factor Theorem states that if x = a is a zero of a polynomial function f(x), then (x - a) is a factor of f(x). In this case, x = 2 is a zero, so we know that (x - 2) is a factor of f(x).
To obtain the other factors, we can perform polynomial long division or use synthetic division to divide f(x) by (x - 2).
Performing synthetic division or polynomial long division, we find that (x - 2) divides f(x) without leaving any remainder, resulting in the quotient g(x) = x^2 + x - 12.
Now we have factored f(x) as (x - 2)(x^2 + x - 12).
To further factor the quadratic factor x^2 + x - 12, we can use factoring or quadratic formula. In this case, (x^2 + x - 12) can be factored as (x + 4)(x - 3).
Therefore, the factored form of f(x) = x^3 - 3x^2 - 10x + 24 is (x - 2)(x + 4)(x - 3).