## Intermediate Algebra (12th Edition)

$\left\{ 0,2,4 \right\}$
$\bf{\text{Solution Outline:}}$ Express the given equation, $x^3-6x^2=-8x ,$ in factored form. Then use the Zero Product Property by equating each factor to zero. Finally, solve each of the resulting equations. $\bf{\text{Solution Details:}}$ Using the properties of equality, the given equation is equivalent to \begin{array}{l}\require{cancel} x^3-6x^2+8x=0 .\end{array} The $GCF$ of the constants of the terms $\{ 1,-6,8 \}$ is $1$ since it is the highest number that can divide all the given constants. The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{ x^3,x^2,x \}$ is $x .$ Hence, the entire expression has $GCF= x .$ Factoring the $GCF= x ,$ the expression above is equivalent to \begin{array}{l}\require{cancel} x(x^2-6x+8)=0 .\end{array} Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $1(8)=8$ and the value of $b$ is $-6 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ -2,-4 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} x(x^2-2x-4x+8)=0 .\end{array} Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to \begin{array}{l}\require{cancel} x[(x^2-2x)-(4x-8)]=0 .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} x[x(x-2)-4(x-2)]=0 .\end{array} Factoring the $GCF= (x-2)$ of the entire expression above results to \begin{array}{l}\require{cancel} x[(x-2)(x-4)]=0 \\\\ x(x-2)(x-4)=0 .\end{array} Equating each factor zero (Zero Product Property), then \begin{array}{l}\require{cancel} x=0 \\\\\text{OR}\\\\ x-2=0 \\\\\text{OR}\\\\ x-4=0 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} x=0 \\\\\text{OR}\\\\ x-2=0 \\\\ x=2 \\\\\text{OR}\\\\ x-4=0 \\\\ x=4 .\end{array} Hence, the solutions are $\left\{ 0,2,4 \right\} .$