Solve a^{2}b^{3}-n^{4}+a^{2}b^{3}x^{2}-n^{4}x^2-3a^2b^3x+3n^4x

$$a ^ { 2 } b ^ { 3 } - n ^ { 4 } + a ^ { 2 } b ^ { 3 } x ^ { 2 } - n ^ { 4 } x ^ { 2 } - 3 a ^ { 2 } b ^ { 3 } x + 3 n ^ { 4 } x$$

Factor

$$\left(x^{2}-3x+1\right)\left(a^{2}b^{3}-n^{4}\right)$$

Solution Steps

Do the grouping a^{2}b^{3}-n^{4}+a^{2}b^{3}x^{2}-n^{4}x^{2}-3a^{2}b^{3}x+3n^{4}x=\left(-n^{4}x^{2}+3n^{4}x-n^{4}\right)+\left(a^{2}b^{3}x^{2}-3a^{2}b^{3}x+a^{2}b^{3}\right), and factor out -n^{4} in the first and a^{2}b^{3} in the second group.

$$-n^{4}\left(x^{2}-3x+1\right)+a^{2}b^{3}\left(x^{2}-3x+1\right)$$

Factor out common term x^{2}-3x+1 by using distributive property. Polynomial x^{2}-3x+1 is not factored since it does not have any rational roots.

$$\left(x^{2}-3x+1\right)\left(-n^{4}+a^{2}b^{3}\right)$$

Evaluate

$$\left(x^{2}-3x+1\right)\left(a^{2}b^{3}-n^{4}\right)$$