Solve x^4+7x^3-5x^2-75x?

$$x ^ { 4 } + 7 x ^ { 3 } - 5 x ^ { 2 } - 75 x ?$$

Factor

$$x\left(x-3\right)\left(x+5\right)^{2}$$

Solution Steps

Factor out x.

$$x\left(x^{3}+7x^{2}-5x-75\right)$$

Consider x^{3}+7x^{2}-5x-75. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -75 and q divides the leading coefficient 1. One such root is -5. Factor the polynomial by dividing it by x+5.

$$\left(x+5\right)\left(x^{2}+2x-15\right)$$

Consider x^{2}+2x-15. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-15\right)=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 2.

$$a=-3 b=5$$

Rewrite x^{2}+2x-15 as \left(x^{2}-3x\right)+\left(5x-15\right).

$$\left(x^{2}-3x\right)+\left(5x-15\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(x-3\right)+5\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x+5\right)$$

Rewrite the complete factored expression.

$$x\left(x+5\right)^{2}\left(x-3\right)$$

Evaluate

$$x\left(x-3\right)\left(x+5\right)^{2}$$