Solve x^4-75x^2+250x

$$x ^ { 4 } - 75 x ^ { 2 } + 250 x$$

Factor

$$x\left(x+10\right)\left(x-5\right)^{2}$$

Solution Steps

Factor out x.

$$x\left(x^{3}-75x+250\right)$$

Consider x^{3}-75x+250. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 250 and q divides the leading coefficient 1. One such root is -10. Factor the polynomial by dividing it by x+10.

$$\left(x+10\right)\left(x^{2}-10x+25\right)$$

Consider x^{2}-10x+25. Use the perfect square formula, a^{2}-2ab+b^{2}=\left(a-b\right)^{2}, where a=x and b=5.

$$\left(x-5\right)^{2}$$

Rewrite the complete factored expression.

$$x\left(x+10\right)\left(x-5\right)^{2}$$

Evaluate

$$x\left(x+10\right)\left(x-5\right)^{2}$$