Solve 4v^5-14v^4+10v^3

$$4 v ^ { 5 } - 14 v ^ { 4 } + 10 v ^ { 3 }$$

Factor

$$2\left(v-1\right)\left(2v-5\right)v^{3}$$

Solution Steps

Factor out 2.

$$2\left(2v^{5}-7v^{4}+5v^{3}\right)$$

Consider 2v^{5}-7v^{4}+5v^{3}. Factor out v^{3}.

$$v^{3}\left(2v^{2}-7v+5\right)$$

Consider 2v^{2}-7v+5. Factor the expression by grouping. First, the expression needs to be rewritten as 2v^{2}+av+bv+5. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\times 5=10$$

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 2v^{2}-7v+5 as \left(2v^{2}-5v\right)+\left(-2v+5\right).

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

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

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

Factor out common term 2v-5 by using distributive property.

$$\left(2v-5\right)\left(v-1\right)$$

Rewrite the complete factored expression.

$$2v^{3}\left(2v-5\right)\left(v-1\right)$$

Evaluate

$$2\left(v-1\right)\left(2v-5\right)v^{3}$$