Solve x^3+9x^2+26x+24+x^3+7x^2+12x+{x}^3+5{x}^2+4x+{x}^3+3{x}^2+2x

$${ x }^{ 3 } +9 { x }^{ 2 } +26x+24+ { x }^{ 3 } +7 { x }^{ 2 } +12x+ { x }^{ 3 } +5 { x }^{ 2 } +4x+ { x }^{ 3 } +3 { x }^{ 2 } +2x$$

Evaluate

$$4\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Solution Steps

Combine x^{3} and x^{3} to get 2x^{3}.

$$2x^{3}+9x^{2}+26x+24+7x^{2}+12x+x^{3}+5x^{2}+4x+x^{3}+3x^{2}+2x$$

Combine 9x^{2} and 7x^{2} to get 16x^{2}.

$$2x^{3}+16x^{2}+26x+24+12x+x^{3}+5x^{2}+4x+x^{3}+3x^{2}+2x$$

Combine 26x and 12x to get 38x.

$$2x^{3}+16x^{2}+38x+24+x^{3}+5x^{2}+4x+x^{3}+3x^{2}+2x$$

Combine 2x^{3} and x^{3} to get 3x^{3}.

$$3x^{3}+16x^{2}+38x+24+5x^{2}+4x+x^{3}+3x^{2}+2x$$

Combine 16x^{2} and 5x^{2} to get 21x^{2}.

$$3x^{3}+21x^{2}+38x+24+4x+x^{3}+3x^{2}+2x$$

Combine 38x and 4x to get 42x.

$$3x^{3}+21x^{2}+42x+24+x^{3}+3x^{2}+2x$$

Combine 3x^{3} and x^{3} to get 4x^{3}.

$$4x^{3}+21x^{2}+42x+24+3x^{2}+2x$$

Combine 21x^{2} and 3x^{2} to get 24x^{2}.

$$4x^{3}+24x^{2}+42x+24+2x$$

Combine 42x and 2x to get 44x.

$$4x^{3}+24x^{2}+44x+24$$

Factor

$$4\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Solution Steps

Multiply and combine like terms.

$$4x^{3}+24x^{2}+44x+24$$

Factor out 4.

$$4\left(x^{3}+6x^{2}+11x+6\right)$$

Consider x^{3}+6x^{2}+11x+6. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 6 and q divides the leading coefficient 1. One such root is -3. Factor the polynomial by dividing it by x+3.

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

Consider x^{2}+3x+2. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=2$$

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

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

Factor out x in the first and 2 in the second group.

$$x\left(x+1\right)+2\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+2\right)$$

Rewrite the complete factored expression.

$$4\left(x+3\right)\left(x+1\right)\left(x+2\right)$$