Solve 3x^9+27x^8+54x^7

$$3 x ^ { 9 } + 27 x ^ { 8 } + 54 x ^ { 7 }$$

Factor

$$3\left(x+3\right)\left(x+6\right)x^{7}$$

Solution Steps

Factor out 3.

$$3\left(x^{9}+9x^{8}+18x^{7}\right)$$

Consider x^{9}+9x^{8}+18x^{7}. Factor out x^{7}.

$$x^{7}\left(x^{2}+9x+18\right)$$

Consider x^{2}+9x+18. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+18. To find a and b, set up a system to be solved.

$$a+b=9 ab=1\times 18=18$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 18.

$$1,18 2,9 3,6$$

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 9.

$$a=3 b=6$$

Rewrite x^{2}+9x+18 as \left(x^{2}+3x\right)+\left(6x+18\right).

$$\left(x^{2}+3x\right)+\left(6x+18\right)$$

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

$$x\left(x+3\right)+6\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+6\right)$$

Rewrite the complete factored expression.

$$3x^{7}\left(x+3\right)\left(x+6\right)$$

Evaluate

$$3\left(x+3\right)\left(x+6\right)x^{7}$$