Solve 3x^3-14x^2+13x+6

$$3 { x }^{ 3 } -14 { x }^{ 2 } +13x+6$$

Factor

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

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 6 and q divides the leading coefficient 3. One such root is 2. Factor the polynomial by dividing it by x-2.

$$\left(x-2\right)\left(3x^{2}-8x-3\right)$$

Consider 3x^{2}-8x-3. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-8 ab=3\left(-3\right)=-9$$

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 -9.

$$1,-9 3,-3$$

Calculate the sum for each pair.

$$1-9=-8 3-3=0$$

The solution is the pair that gives sum -8.

$$a=-9 b=1$$

Rewrite 3x^{2}-8x-3 as \left(3x^{2}-9x\right)+\left(x-3\right).

$$\left(3x^{2}-9x\right)+\left(x-3\right)$$

Factor out 3x in 3x^{2}-9x.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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