Solve 16x^4-25x^2y^2+9y^4

$$16 x ^ { 4 } - 25 x ^ { 2 } y ^ { 2 } + 9 y ^ { 4 }$$

Factor

$$\left(x-y\right)\left(4x-3y\right)\left(x+y\right)\left(4x+3y\right)$$

Solution Steps

Consider 16x^{4}-25x^{2}y^{2}+9y^{4} as a polynomial over variable x.

$$16x^{4}-25y^{2}x^{2}+9y^{4}$$

Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 16x^{4} and n divides the constant factor 9y^{4}. One such factor is 16x^{2}-9y^{2}. Factor the polynomial by dividing it by this factor.

$$\left(16x^{2}-9y^{2}\right)\left(x^{2}-y^{2}\right)$$

Consider 16x^{2}-9y^{2}. Rewrite 16x^{2}-9y^{2} as \left(4x\right)^{2}-\left(3y\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(4x-3y\right)\left(4x+3y\right)$$

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

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

Rewrite the complete factored expression.

$$\left(x-y\right)\left(x+y\right)\left(4x-3y\right)\left(4x+3y\right)$$

Evaluate

$$16x^{4}+9y^{4}-25\left(xy\right)^{2}$$