Question

Use an appropriate procedure to show that (x-2) is a factor of the function f(x)=x^5-4x^4+3x^3-x^2+12 ?

Answer 1

Please see below.

Explanation:

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`f(x)=x^5-4x^4+3x^3-x^2+12`

`f(x)=x^5-2x^4-2x^4+4x^3-x^3+2x^2-3x^2+12`

`f(x)=x^4(x-2)-2x^3(x-2)-x^2(x-2)-3(x^2-4)`

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

`f(x)=x^4(x-2)-2x^3(x-2)-x^2(x-2)-(3x+6)(x-2)`

Now, we can factor `(x-2)` out:

`f(x)=(x-2)(x^4-2x^3-x^2-3x-6)`

You can also solve this problem by performing a long division of `f(x)` by `x-2`.