Solve a^{4}-2a^{3}b-8a^2b^2-6ab^3-b^4

$$a ^ { 4 } - 2 a ^ { 3 } b - 8 a ^ { 2 } b ^ { 2 } - 6 a b ^ { 3 } - b ^ { 4 }$$

Factor

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

Solution Steps

Consider a^{4}-2a^{3}b-8a^{2}b^{2}-6ab^{3}-b^{4} as a polynomial over variable a.

$$a^{4}-2ba^{3}-8b^{2}a^{2}-6b^{3}a-b^{4}$$

Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{4} and m divides the constant factor -b^{4}. One such factor is a+b. Factor the polynomial by dividing it by this factor.

$$\left(a+b\right)\left(a^{3}-5ab^{2}-b^{3}-3ba^{2}\right)$$

Consider a^{3}-5ab^{2}-b^{3}-3ba^{2}. Consider a^{3}-5ab^{2}-b^{3}-3ba^{2} as a polynomial over variable a.

$$a^{3}-3ba^{2}-5b^{2}a-b^{3}$$

Find one factor of the form a^{n}+p, where a^{n} divides the monomial with the highest power a^{3} and p divides the constant factor -b^{3}. One such factor is a+b. Factor the polynomial by dividing it by this factor.

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

Rewrite the complete factored expression.

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

Evaluate

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