Solve 3x^3+31x^2+82x+24

$$3 x ^ { 3 } + 31 x ^ { 2 } + 82 x + 24$$

Factor

$$\left(x+4\right)\left(x+6\right)\left(3x+1\right)$$

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 24 and q divides the leading coefficient 3. One such root is -4. Factor the polynomial by dividing it by x+4.

$$\left(x+4\right)\left(3x^{2}+19x+6\right)$$

Consider 3x^{2}+19x+6. Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=19 ab=3\times 6=18$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 18.

$$1,18 2,9 3,6$$

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 19.

$$a=1 b=18$$

Rewrite 3x^{2}+19x+6 as \left(3x^{2}+x\right)+\left(18x+6\right).

$$\left(3x^{2}+x\right)+\left(18x+6\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(3x+1\right)+6\left(3x+1\right)$$

Factor out common term 3x+1 by using distributive property.

$$\left(3x+1\right)\left(x+6\right)$$

Rewrite the complete factored expression.

$$\left(3x+1\right)\left(x+4\right)\left(x+6\right)$$

Evaluate

$$\left(x+4\right)\left(x+6\right)\left(3x+1\right)$$