Solve x^4-2x^3-35x^2

$$x ^ { 4 } - 2 x ^ { 3 } - 35 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out x^{2}.

$$x^{2}\left(x^{2}-2x-35\right)$$

Consider x^{2}-2x-35. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-35. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-35\right)=-35$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -35.

$$1,-35 5,-7$$

Calculate the sum for each pair.

$$1-35=-34 5-7=-2$$

The solution is the pair that gives sum -2.

$$a=-7 b=5$$

Rewrite x^{2}-2x-35 as \left(x^{2}-7x\right)+\left(5x-35\right).

$$\left(x^{2}-7x\right)+\left(5x-35\right)$$

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

$$x\left(x-7\right)+5\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(x+5\right)$$

Rewrite the complete factored expression.

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

Evaluate

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