Answer: C. 4
Explanation:
To find the x-intercepts of the function f(x)=x3−5x2−8x+48f(x)=x3−5x2−8x+48, set f(x) equal to zero and solve for x, x:x3−5x2−8x+48=0x3−5x2−8x+48=0 You can use various methods to solve this cubic equation, such as factoring, synthetic division, or numerical methods. Let's use a numerical method, such as the Newton-Raphson method, to find one root to make it easier. The synthetic division suggests that x=4x=4 is a root:f(4)=43−5(4)2−8(4)+48f(4)=43−5(4)2−8(4)+48=64−80−32+48=64−80−32+48=0=0Since x=4x=4 makes the function equal to zero, (x−4)(x−4) is a factor. Now, we can perform synthetic division to find the quadratic factor:41−5−8481−100411−5−1−80480The result is x2−xx2−x. So, the factored form is:f(x)=(x−4)(x2−x)f(x)=(x−4)(x2−x)Now, set each factor equal to zero to find the x-intercepts:x−4=0x−4=0 gives x=4x=4.x2−x=0x2−x=0 gives x(x−1)=0x(x−1)=0, so x=0x=0 and x=1x=1.Therefore, the x-intercepts of the function f(x)f(x) are x=0x=0, x=1x=1, and x=4x=4.