Solve 9t^3-90t^2+144t

$$9 t ^ { 3 } - 90 t ^ { 2 } + 144 t$$

Factor

$$9t\left(t-8\right)\left(t-2\right)$$

Solution Steps

Factor out 9.

$$9\left(t^{3}-10t^{2}+16t\right)$$

Consider t^{3}-10t^{2}+16t. Factor out t.

$$t\left(t^{2}-10t+16\right)$$

Consider t^{2}-10t+16. Factor the expression by grouping. First, the expression needs to be rewritten as t^{2}+at+bt+16. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\times 16=16$$

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 16.

$$-1,-16 -2,-8 -4,-4$$

Calculate the sum for each pair.

$$-1-16=-17 -2-8=-10 -4-4=-8$$

The solution is the pair that gives sum -10.

$$a=-8 b=-2$$

Rewrite t^{2}-10t+16 as \left(t^{2}-8t\right)+\left(-2t+16\right).

$$\left(t^{2}-8t\right)+\left(-2t+16\right)$$

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

$$t\left(t-8\right)-2\left(t-8\right)$$

Factor out common term t-8 by using distributive property.

$$\left(t-8\right)\left(t-2\right)$$

Rewrite the complete factored expression.

$$9t\left(t-8\right)\left(t-2\right)$$

Evaluate

$$9t\left(t-8\right)\left(t-2\right)$$