Solve a^4-b^4+a^4b^4-1

$$a ^ { 4 } - b ^ { 4 } + a ^ { 4 } b ^ { 4 } - 1$$

Factor

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

Solution Steps

Do the grouping a^{4}-b^{4}+a^{4}b^{4}-1=\left(a^{4}b^{4}+a^{4}\right)+\left(-b^{4}-1\right), and factor out a^{4} in the first and -1 in the second group.

$$a^{4}\left(b^{4}+1\right)-\left(b^{4}+1\right)$$

Factor out common term b^{4}+1 by using distributive property.

$$\left(b^{4}+1\right)\left(a^{4}-1\right)$$

Consider a^{4}-1. Rewrite a^{4}-1 as \left(a^{2}\right)^{2}-1^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(a^{2}-1\right)\left(a^{2}+1\right)$$

Consider a^{2}-1. Rewrite a^{2}-1 as a^{2}-1^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(a-1\right)\left(a+1\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: a^{2}+1,b^{4}+1.

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

Evaluate

$$\left(a^{4}-1\right)\left(b^{4}+1\right)$$