Solve 7n^3-35n^2+28n

$$7 n ^ { 3 } - 35 n ^ { 2 } + 28 n$$

Factor

$$7n\left(n-4\right)\left(n-1\right)$$

Solution Steps

Factor out 7.

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

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

$$n\left(n^{2}-5n+4\right)$$

Consider n^{2}-5n+4. Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn+4. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\times 4=4$$

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 4.

$$-1,-4 -2,-2$$

Calculate the sum for each pair.

$$-1-4=-5 -2-2=-4$$

The solution is the pair that gives sum -5.

$$a=-4 b=-1$$

Rewrite n^{2}-5n+4 as \left(n^{2}-4n\right)+\left(-n+4\right).

$$\left(n^{2}-4n\right)+\left(-n+4\right)$$

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

$$n\left(n-4\right)-\left(n-4\right)$$

Factor out common term n-4 by using distributive property.

$$\left(n-4\right)\left(n-1\right)$$

Rewrite the complete factored expression.

$$7n\left(n-4\right)\left(n-1\right)$$

Evaluate

$$7n\left(n-4\right)\left(n-1\right)$$