Chapter A.10, Problem 105AYU

To determine

Factor the expression $3{\left(x^2+4\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+x.4{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}.2x$ and express the answer so that only positive exponents occur.

Answer to Problem 105AYU

The factorized form of the expression $3{\left(x^2+4\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+x.4{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}.2x$ is ${\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}.\left(11 x^2+12\right)$.

Explanation of Solution

Given:

It is asked to determine the factor of each expression and express the answer that contains only positive exponents.

Calculation:

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

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

$={\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left[3\left(x^2+4\right)+8x^2\right]$

$={\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}.\left(11 x^2+12\right)$