Solve 3x^4-14x^3-5x^2

$$3 x ^ { 4 } - 14 x ^ { 3 } - 5 x ^ { 2 }$$

Factor

$$\left(x-5\right)\left(3x+1\right)x^{2}$$

Solution Steps

Factor out x^{2}.

$$x^{2}\left(3x^{2}-14x-5\right)$$

Consider 3x^{2}-14x-5. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=-14 ab=3\left(-5\right)=-15$$

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 -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 -14.

$$a=-15 b=1$$

Rewrite 3x^{2}-14x-5 as \left(3x^{2}-15x\right)+\left(x-5\right).

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

Factor out 3x in 3x^{2}-15x.

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

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(3x+1\right)$$

Rewrite the complete factored expression.

$$x^{2}\left(x-5\right)\left(3x+1\right)$$

Evaluate

$$\left(x-5\right)\left(3x+1\right)x^{2}$$