Solve d^4-2d^2c^2+c^4

$$d ^ { 4 } - 2 d ^ { 2 } c ^ { 2 } + c ^ { 4 }$$

Factor

$$\left(d-c\right)^{2}\left(c+d\right)^{2}$$

Solution Steps

Consider d^{4}-2d^{2}c^{2}+c^{4} as a polynomial over variable d.

$$d^{4}-2c^{2}d^{2}+c^{4}$$

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

$$\left(-c^{2}+d^{2}\right)\left(-c^{2}+d^{2}\right)$$

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

$$\left(d-c\right)\left(d+c\right)$$

Reorder the terms.

$$\left(-c+d\right)\left(c+d\right)$$

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

$$\left(d-c\right)\left(d+c\right)$$

Reorder the terms.

$$\left(-c+d\right)\left(c+d\right)$$

Rewrite the complete factored expression.

$$\left(-c+d\right)^{2}\left(c+d\right)^{2}$$

Evaluate

$$\left(d^{2}-c^{2}\right)^{2}$$