Question

Factor the expression completely.

${\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

Short answer

The solution${\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=\frac{x^2+9}{3{\left(x^2+3\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

Step-by-step solution

Step 1. Apply the concept of distributive property.

If A, B and C are any real numbers or algebraic expressions, then

$A\left(B+C\right)=\left(B+C\right)A=AB+AC$

Step 2. Analyze the polynomial.

Given${\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

Step 3. Find the greatest common factor.

${\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$is the polynomial

Take the common factor ${\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$out .

Step 4. Factor the polynomial.

Now, the factors of polynomial${\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$are given by:

$\begin{array}{l}{\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}={\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(1-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-1}\right)\\ \Rightarrow {\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=\frac{1}{{\left(x^2+3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\left(1-\frac{2}{3}{x}^2\frac{1}{\left(x^2+3\right)}\right)\mathrm{...........}\left[a^{-n}=\frac{1}{a^n}\right]\\ \Rightarrow {\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=\frac{1}{{\left(x^2+3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\left(1\cdot \frac{3\left(x^2+3\right)}{3\left(x^2+3\right)}-\frac{2x^2}{3\left(x^2+3\right)}\right)\\ \Rightarrow {\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=\frac{1}{{\left(x^2+3\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\left(\frac{3x^2+9-2x^2}{3\left(x^2+3\right)}\right)\\ \Rightarrow {\left(x^2+3\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-\frac{2}{3}{x}^2{\left(x^2+3\right)}^{-\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=\frac{x^2+9}{3{\left(x^2+3\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\mathrm{.........................}\left[x^a\cdot x^b=x^{a+b}\right]\end{array}$