Solve 18h^3+45h^2-8h-20

$$18 { h }^{ 3 } +45 { h }^{ 2 } -8h-20$$

Factor

$$\left(3h-2\right)\left(2h+5\right)\left(3h+2\right)$$

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 -20 and q divides the leading coefficient 18. One such root is \frac{2}{3}. Factor the polynomial by dividing it by 3h-2.

$$\left(3h-2\right)\left(6h^{2}+19h+10\right)$$

Consider 6h^{2}+19h+10. Factor the expression by grouping. First, the expression needs to be rewritten as 6h^{2}+ah+bh+10. To find a and b, set up a system to be solved.

$$a+b=19 ab=6\times 10=60$$

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 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Calculate the sum for each pair.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

The solution is the pair that gives sum 19.

$$a=4 b=15$$

Rewrite 6h^{2}+19h+10 as \left(6h^{2}+4h\right)+\left(15h+10\right).

$$\left(6h^{2}+4h\right)+\left(15h+10\right)$$

Factor out 2h in the first and 5 in the second group.

$$2h\left(3h+2\right)+5\left(3h+2\right)$$

Factor out common term 3h+2 by using distributive property.

$$\left(3h+2\right)\left(2h+5\right)$$

Rewrite the complete factored expression.

$$\left(3h-2\right)\left(3h+2\right)\left(2h+5\right)$$

Evaluate

$$\left(2h+5\right)\left(9h^{2}-4\right)$$