Solve 35m^4-375m^3+250m^2

$$35 m ^ { 4 } - 375 m ^ { 3 } + 250 m ^ { 2 }$$

Factor

$$5\left(m-10\right)\left(7m-5\right)m^{2}$$

Solution Steps

Factor out 5.

$$5\left(7m^{4}-75m^{3}+50m^{2}\right)$$

Consider 7m^{4}-75m^{3}+50m^{2}. Factor out m^{2}.

$$m^{2}\left(7m^{2}-75m+50\right)$$

Consider 7m^{2}-75m+50. Factor the expression by grouping. First, the expression needs to be rewritten as 7m^{2}+am+bm+50. To find a and b, set up a system to be solved.

$$a+b=-75 ab=7\times 50=350$$

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 350.

$$-1,-350 -2,-175 -5,-70 -7,-50 -10,-35 -14,-25$$

Calculate the sum for each pair.

$$-1-350=-351 -2-175=-177 -5-70=-75 -7-50=-57 -10-35=-45 -14-25=-39$$

The solution is the pair that gives sum -75.

$$a=-70 b=-5$$

Rewrite 7m^{2}-75m+50 as \left(7m^{2}-70m\right)+\left(-5m+50\right).

$$\left(7m^{2}-70m\right)+\left(-5m+50\right)$$

Factor out 7m in the first and -5 in the second group.

$$7m\left(m-10\right)-5\left(m-10\right)$$

Factor out common term m-10 by using distributive property.

$$\left(m-10\right)\left(7m-5\right)$$

Rewrite the complete factored expression.

$$5m^{2}\left(m-10\right)\left(7m-5\right)$$

Evaluate

$$5\left(m-10\right)\left(7m-5\right)m^{2}$$