Question

If f(x)=x^3-12x^2+35x-24f(x)=x 3 −12x 2 +35x−24 and f(8)=0f(8)=0, then find all of the zeros of f(x)f(x) algebraically.

Answer 1

The zeros of f(x) are: (x - 1), (x - 3) and (x - 8)

Explanation:

Given

`f(x)=x^3-12x^2+35x-24`

`f(8) = 0`

Required

Find all zeros of the f(x)

If
`f(8) = 0` then:

`x = 8`

And
`x - 8` is a factor

Divide f(x) by x - 8

`(f(x))/(x - 8) = (x^3-12x^2+35x-24)/(x - 8)`

Expand the numerator

`(f(x))/(x - 8) = (x^3 - 4x^2 -8x^2 + 3x + 32x - 24)/(x - 8)`

Rewrite as:

`(f(x))/(x - 8) = (x^3 - 4x^2 + 3x - 8x^2 +32x - 24)/(x - 8)`

Factorize

`(f(x))/(x - 8) = ((x^2 - 4x + 3)(x - 8))/(x - 8)`

Expand

`(f(x))/(x - 8) = ((x^2 -x - 3x + 3)(x - 8))/(x - 8)`

Factorize

`(f(x))/(x - 8) = ((x - 1)(x - 3)(x - 8))/(x - 8)`

`(f(x))/(x - 8) = (x - 1)(x - 3)`

Multiply both sides by x - 8

`f(x) = (x - 1)(x - 3)(x - 8)`

Hence, the zeros of f(x) are: (x - 1), (x - 3) and (x - 8)