Solve 2p^5+10p^4-12p^3

$$2 p ^ { 5 } + 10 p ^ { 4 } - 12 p ^ { 3 }$$

Factor

$$2\left(p-1\right)\left(p+6\right)p^{3}$$

Solution Steps

Factor out 2.

$$2\left(p^{5}+5p^{4}-6p^{3}\right)$$

Consider p^{5}+5p^{4}-6p^{3}. Factor out p^{3}.

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

Consider p^{2}+5p-6. Factor the expression by grouping. First, the expression needs to be rewritten as p^{2}+ap+bp-6. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\left(-6\right)=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$a=-1 b=6$$

Rewrite p^{2}+5p-6 as \left(p^{2}-p\right)+\left(6p-6\right).

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

Factor out p in the first and 6 in the second group.

$$p\left(p-1\right)+6\left(p-1\right)$$

Factor out common term p-1 by using distributive property.

$$\left(p-1\right)\left(p+6\right)$$

Rewrite the complete factored expression.

$$2p^{3}\left(p-1\right)\left(p+6\right)$$

Evaluate

$$2\left(p-1\right)\left(p+6\right)p^{3}$$