Solve 3x^4-5x^3-2x^2

$$3 { x }^{ 4 } -5 { x }^{ 3 } -2 { x }^{ 2 }$$

Factor

$$\left(x-2\right)\left(3x+1\right)x^{2}$$

Solution Steps

Factor out x^{2}.

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

Consider 3x^{2}-5x-2. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-5 ab=3\left(-2\right)=-6$$

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 -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -5.

$$a=-6 b=1$$

Rewrite 3x^{2}-5x-2 as \left(3x^{2}-6x\right)+\left(x-2\right).

$$\left(3x^{2}-6x\right)+\left(x-2\right)$$

Factor out 3x in 3x^{2}-6x.

$$3x\left(x-2\right)+x-2$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(3x+1\right)$$

Rewrite the complete factored expression.

$$x^{2}\left(x-2\right)\left(3x+1\right)$$

Evaluate

$$\left(x-2\right)\left(3x+1\right)x^{2}$$