Solve -d^3+d^2+30d

$$- d ^ { 3 } + d ^ { 2 } + 30 d$$

Factor

$$-d\left(d-6\right)\left(d+5\right)$$

Solution Steps

Factor out d.

$$d\left(-d^{2}+d+30\right)$$

Consider -d^{2}+d+30. Factor the expression by grouping. First, the expression needs to be rewritten as -d^{2}+ad+bd+30. To find a and b, set up a system to be solved.

$$a+b=1 ab=-30=-30$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -30.

$$-1,30 -2,15 -3,10 -5,6$$

Calculate the sum for each pair.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

The solution is the pair that gives sum 1.

$$a=6 b=-5$$

Rewrite -d^{2}+d+30 as \left(-d^{2}+6d\right)+\left(-5d+30\right).

$$\left(-d^{2}+6d\right)+\left(-5d+30\right)$$

Factor out -d in the first and -5 in the second group.

$$-d\left(d-6\right)-5\left(d-6\right)$$

Factor out common term d-6 by using distributive property.

$$\left(d-6\right)\left(-d-5\right)$$

Rewrite the complete factored expression.

$$d\left(d-6\right)\left(-d-5\right)$$

Evaluate

$$-d\left(d-6\right)\left(d+5\right)$$