Solve ac^4-c^4-25ac^2+25c^2

$$a c ^ { 4 } - c ^ { 4 } - 25 a c ^ { 2 } + 25 c ^ { 2 }$$

Factor

$$\left(a-1\right)\left(c-5\right)\left(c+5\right)c^{2}$$

Solution Steps

Factor out c^{2}.

$$c^{2}\left(ac^{2}-c^{2}-25a+25\right)$$

Consider ac^{2}-c^{2}-25a+25. Do the grouping ac^{2}-c^{2}-25a+25=\left(ac^{2}-c^{2}\right)+\left(-25a+25\right), and factor out c^{2} in the first and -25 in the second group.

$$c^{2}\left(a-1\right)-25\left(a-1\right)$$

Factor out common term a-1 by using distributive property.

$$\left(a-1\right)\left(c^{2}-25\right)$$

Consider c^{2}-25. Rewrite c^{2}-25 as c^{2}-5^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(c-5\right)\left(c+5\right)$$

Rewrite the complete factored expression.

$$c^{2}\left(a-1\right)\left(c-5\right)\left(c+5\right)$$

Evaluate

$$\left(a-1\right)c^{2}\left(c^{2}-25\right)$$