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Perfect Square Trinomials

Find the GCD ...

Question

Find the GCD of the following polynominals $3 x^{4} + 6 x^{3} - 12 x^{2} - 24 x$ and $4 x^{4} + x^{3} + 8 x^{2} - 8 x$ .

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Solution

Let $f (x) = 3 x^{4} + 6 x^{3} - 12 x^{2} - 24 x = 3 x (x^{3} + 2 x^{2} - 4 x - 8)$ .
Let $g (x) = 4 x^{4} + 14 x^{3} + 8 x^{2} - 8 x = 2 x (2 x^{3} + 7 x^{2} + 4 x - 4)$
LetusfindtheGCDforthepolynomials $x^{3} + 2 x^{2} - 4 x - 8$ and $2 x^{3} + 7 x^{2} + 4 x - 4$
We choose the divisor to be $x^{3} + 2 x^{2} - 4 x - 8$ .
Common factor of $x^{3} + 2 x^{2} - 4 x - 8$ and $2 x^{3} + 7 x^{2} + 4 x - 4$ is $x^{2} + 4 x + 4$
Also common factor of 3x and 2x is x.Thus,GCD $(f (x), g (x)) = x (x^{2} + 4 x + 4)$ .

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