Question

How do you divide $\dfrac{{{x}^{4}}-16}{x+2}$?

Answer 1

Hint: The given quadratic equations need to be simplified. The equation can be factored into the multiplication form of linear equations. If there is any common factor then we eliminate that to find the simplified form of the given expression $\dfrac{{{x}^{4}}-16}{x+2}$.

Complete step by step solution:
We have been given the division of one quadratic equation and one linear equation. We need to find the factor of the quadratic equation.
The quadratic equation is $\left( {{x}^{4}}-16 \right)$.
For $g\left( x \right)=\left( {{x}^{4}}-16 \right)$, we use the identity ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$.
Putting the values of $a={{x}^{2}};b=4$, we get \[{{x}^{4}}-16=\left( {{x}^{2}}-4 \right)\left( {{x}^{2}}+4 \right)\].
We can again factor the quadratic equation \[\left( {{x}^{2}}-4 \right)\] using same process.
We take the values of $a=x;b=2$ for ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$.
So, \[\left( {{x}^{2}}-4 \right)=\left( x-2 \right)\left( x+2 \right)\].
So, final factorisation is \[{{x}^{4}}-16=\left( x-2 \right)\left( x+2 \right)\left( {{x}^{2}}+4 \right)\]
Now we put the factorised values for the two equations.
We get \[\dfrac{{{x}^{4}}-16}{x+2}=\dfrac{\left( x-2 \right)\left( x+2 \right)\left( {{x}^{2}}+4 \right)}{\left( x+2 \right)}\].
Now we can see in the denominator and the numerator the common factor is \[\left( x+2 \right)\].
We can eliminate the factor from both denominator and the numerator.
So, \[\dfrac{{{x}^{4}}-16}{x+2}=\dfrac{\left( x-2 \right)\left( x+2 \right)\left( {{x}^{2}}+4 \right)}{\left( x+2 \right)}=\left( x-2 \right)\left( {{x}^{2}}+4 \right)\]. The only condition being $x\ne -2$.
Therefore, the simplified form of \[\dfrac{{{x}^{4}}-16}{x+2}\] is \[\left( x-2 \right)\left( {{x}^{2}}+4 \right)\].

Note: We can verify the factors for the functions.
We can validate the other solution of $x=2$ for $f\left( x \right)=\left( {{x}^{4}}-16 \right)$. We put the value of $x=2$ in the equation.
\[f\left( 2 \right)=\left( {{2}^{4}}-16 \right)=16-16=0\].