Solve y^4+2y^3z-2yz^3-z^4

$$y ^ { 4 } + 2 y ^ { 3 } z - 2 y z ^ { 3 } - z ^ { 4 }$$

Factor

$$\left(y-z\right)\left(y+z\right)^{3}$$

Solution Steps

Consider y^{4}+2y^{3}z-2yz^{3}-z^{4} as a polynomial over variable y.

$$y^{4}+2zy^{3}-2z^{3}y-z^{4}$$

Find one factor of the form y^{k}+m, where y^{k} divides the monomial with the highest power y^{4} and m divides the constant factor -z^{4}. One such factor is y^{2}-z^{2}. Factor the polynomial by dividing it by this factor.

$$\left(y^{2}-z^{2}\right)\left(y^{2}+2yz+z^{2}\right)$$

Consider y^{2}-z^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(y-z\right)\left(y+z\right)$$

Consider y^{2}+2yz+z^{2}. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=y and b=z.

$$\left(y+z\right)^{2}$$

Rewrite the complete factored expression.

$$\left(y-z\right)\left(y+z\right)^{3}$$

Evaluate

$$\left(y-z\right)\left(y+z\right)^{3}$$