Solve 12v^6-44v^5+40v^4

$$12 v ^ { 6 } - 44 v ^ { 5 } + 40 v ^ { 4 }$$

Factor

$$4\left(v-2\right)\left(3v-5\right)v^{4}$$

Solution Steps

Factor out 4.

$$4\left(3v^{6}-11v^{5}+10v^{4}\right)$$

Consider 3v^{6}-11v^{5}+10v^{4}. Factor out v^{4}.

$$v^{4}\left(3v^{2}-11v+10\right)$$

Consider 3v^{2}-11v+10. Factor the expression by grouping. First, the expression needs to be rewritten as 3v^{2}+av+bv+10. To find a and b, set up a system to be solved.

$$a+b=-11 ab=3\times 10=30$$

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 30.

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

Calculate the sum for each pair.

$$-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11$$

The solution is the pair that gives sum -11.

$$a=-6 b=-5$$

Rewrite 3v^{2}-11v+10 as \left(3v^{2}-6v\right)+\left(-5v+10\right).

$$\left(3v^{2}-6v\right)+\left(-5v+10\right)$$

Factor out 3v in the first and -5 in the second group.

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

Factor out common term v-2 by using distributive property.

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

Rewrite the complete factored expression.

$$4v^{4}\left(v-2\right)\left(3v-5\right)$$

Evaluate

$$4\left(v-2\right)\left(3v-5\right)v^{4}$$