Solve q^4-10q^3+21q^2

$$q ^ { 4 } - 10 q ^ { 3 } + 21 q ^ { 2 }$$

Factor

$$\left(q-7\right)\left(q-3\right)q^{2}$$

Solution Steps

Factor out q^{2}.

$$q^{2}\left(q^{2}-10q+21\right)$$

Consider q^{2}-10q+21. Factor the expression by grouping. First, the expression needs to be rewritten as q^{2}+aq+bq+21. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\times 21=21$$

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 21.

$$-1,-21 -3,-7$$

Calculate the sum for each pair.

$$-1-21=-22 -3-7=-10$$

The solution is the pair that gives sum -10.

$$a=-7 b=-3$$

Rewrite q^{2}-10q+21 as \left(q^{2}-7q\right)+\left(-3q+21\right).

$$\left(q^{2}-7q\right)+\left(-3q+21\right)$$

Factor out q in the first and -3 in the second group.

$$q\left(q-7\right)-3\left(q-7\right)$$

Factor out common term q-7 by using distributive property.

$$\left(q-7\right)\left(q-3\right)$$

Rewrite the complete factored expression.

$$q^{2}\left(q-7\right)\left(q-3\right)$$

Evaluate

$$\left(q-7\right)\left(q-3\right)q^{2}$$