Solve 4y^7-26y^6+12y^5

$$4 y ^ { 7 } - 26 y ^ { 6 } + 12 y ^ { 5 }$$

Factor

$$2\left(y-6\right)\left(2y-1\right)y^{5}$$

Solution Steps

Factor out 2.

$$2\left(2y^{7}-13y^{6}+6y^{5}\right)$$

Consider 2y^{7}-13y^{6}+6y^{5}. Factor out y^{5}.

$$y^{5}\left(2y^{2}-13y+6\right)$$

Consider 2y^{2}-13y+6. Factor the expression by grouping. First, the expression needs to be rewritten as 2y^{2}+ay+by+6. To find a and b, set up a system to be solved.

$$a+b=-13 ab=2\times 6=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -13.

$$a=-12 b=-1$$

Rewrite 2y^{2}-13y+6 as \left(2y^{2}-12y\right)+\left(-y+6\right).

$$\left(2y^{2}-12y\right)+\left(-y+6\right)$$

Factor out 2y in the first and -1 in the second group.

$$2y\left(y-6\right)-\left(y-6\right)$$

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

$$\left(y-6\right)\left(2y-1\right)$$

Rewrite the complete factored expression.

$$2y^{5}\left(y-6\right)\left(2y-1\right)$$

Evaluate

$$2\left(y-6\right)\left(2y-1\right)y^{5}$$