Mathematics: Show all work to factor x4 − 5x2 + 4 completely.

Show all work to factor x4 − 5x2 + 4 completely.

kinggonzales5082

06.06.2020

The complete factored form of the given polynomial is (x+1)(x-1)(x+2)(x-2).

Step-by-step explanation: We are given to factorize the following polynomial completely :

P=x^4-5x^2+4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)

We will be using the following formulas :

$(i)~z^2-(a+b)z+ab=(z-a)(z-b),\\\\(ii)~a^2-b^2=(a+b)(a-b).$

Let us consider that

x^2=y.

Then, from (i), we get

$P\\\\=x^4-5x^2+4\\\\=y^2-5y+4\\\\=y^2-(4+1)y+4\\\\=(y-1)(y-4)\\\\=(x^2-1)(x^2-4)\\\\=(x^2-1^2)(x^2-2^2)\\\\=(x+1)(x-1)(x+2)(x-2).$

Thus, the complete factored form of the given polynomial is (x+1)(x-1)(x+2)(x-2).

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