`x^4 - 2x^3 - 2x^2 + 8` is denoted by `f(x)` . It is given that` f(x)` is divisible by `x^2 - 4x + 4` . Prove that `f(x)` is never negative.

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tiburtius | High School Teacher | (Level 2) Educator

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If polynomial `f(x)` is divisible by `x^2-4x+4` then there exist polynomial `g(x)` such that

`f(x)=(x^2-4x+4)cdot g(x)`

We can find `g(x)` by dividing `f(x)` with `x^2-4x+4.`

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

Hence we have

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

We have now written `f(x)` as a product and product is non-negative if both its terms are positive or if both terms are negative or if one of them is 0.

`f(x)=(x-2)^2((x^2+2x+1)+1)=(x-2)^2((x+1)^2+1)`

Now we see that both terms are non-negative.

`(x-2)^2>=0`  ` `because square of any number is non-negative.

`(x+1)^2+1>0`  because `(x+1)^2>=0` and +1 makes it positive.

This concludes our proof.

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n00boxular | eNotes Newbie

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We have now written  as a product and product is non-negative if both its terms are positive or if both terms are negative or if one of them is 0.


So bascially, all you did her was to factorise the 2 quadratic equations to get ` `` `

`(x-2)^2 `

`And`

`(x+1)^2 + 1`

`Right?

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