Solve 2y^3-24y^2+22y

$$2 y ^ { 3 } - 24 y ^ { 2 } + 22 y$$

Factor

$$2y\left(y-11\right)\left(y-1\right)$$

Solution Steps

Factor out 2.

$$2\left(y^{3}-12y^{2}+11y\right)$$

Consider y^{3}-12y^{2}+11y. Factor out y.

$$y\left(y^{2}-12y+11\right)$$

Consider y^{2}-12y+11. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+11. To find a and b, set up a system to be solved.

$$a+b=-12 ab=1\times 11=11$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-11 b=-1$$

Rewrite y^{2}-12y+11 as \left(y^{2}-11y\right)+\left(-y+11\right).

$$\left(y^{2}-11y\right)+\left(-y+11\right)$$

Factor out y in the first and -1 in the second group.

$$y\left(y-11\right)-\left(y-11\right)$$

Factor out common term y-11 by using distributive property.

$$\left(y-11\right)\left(y-1\right)$$

Rewrite the complete factored expression.

$$2y\left(y-11\right)\left(y-1\right)$$

Evaluate

$$2y\left(y-11\right)\left(y-1\right)$$