Solve 3m^3-21m^2+30m

$$3 m ^ { 3 } - 21 m ^ { 2 } + 30 m$$

Factor

$$3m\left(m-5\right)\left(m-2\right)$$

Solution Steps

Factor out 3.

$$3\left(m^{3}-7m^{2}+10m\right)$$

Consider m^{3}-7m^{2}+10m. Factor out m.

$$m\left(m^{2}-7m+10\right)$$

Consider m^{2}-7m+10. Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm+10. To find a and b, set up a system to be solved.

$$a+b=-7 ab=1\times 10=10$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 10.

$$-1,-10 -2,-5$$

Calculate the sum for each pair.

$$-1-10=-11 -2-5=-7$$

The solution is the pair that gives sum -7.

$$a=-5 b=-2$$

Rewrite m^{2}-7m+10 as \left(m^{2}-5m\right)+\left(-2m+10\right).

$$\left(m^{2}-5m\right)+\left(-2m+10\right)$$

Factor out m in the first and -2 in the second group.

$$m\left(m-5\right)-2\left(m-5\right)$$

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

$$\left(m-5\right)\left(m-2\right)$$

Rewrite the complete factored expression.

$$3m\left(m-5\right)\left(m-2\right)$$

Evaluate

$$3m\left(m-5\right)\left(m-2\right)$$