Question

the rectangle below has an area of $14x^{4}+6x^{2}$. the width of the rectangle is equal to the greatest common monomial factor of $14x^{4}$ and $6x^{2}$. what is the length and width of the rectangle? width = length =

Explanation:

Step1: Find the greatest common monomial factor (width)

For the coefficients \(14\) and \(6\), the greatest common factor of \(14\) and \(6\) is \(2\). For the variables \(x^{4}\) and \(x^{2}\), the greatest common factor is \(x^{2}\) (using the rule \(x^{m}\) and \(x^{n}\), \(m\geq n\), \(GCF=x^{n}\)). So the greatest common monomial factor (width) is \(2x^{2}\).

Step2: Find the length

Since the area of a rectangle \(A = length\times width\), and \(A=14x^{4}+6x^{2}\), \(width = 2x^{2}\). Then \(length=\frac{14x^{4}+6x^{2}}{2x^{2}}\).
Using the rule \(\frac{a + b}{c}=\frac{a}{c}+\frac{b}{c}\) (\(c
eq0\)), we have \(\frac{14x^{4}}{2x^{2}}+\frac{6x^{2}}{2x^{2}}\).
For \(\frac{14x^{4}}{2x^{2}}\), using \(\frac{x^{m}}{x^{n}}=x^{m - n}\) (\(m\geq n\)), \(\frac{14x^{4}}{2x^{2}}=7x^{2}\); for \(\frac{6x^{2}}{2x^{2}} = 3\). So \(length=7x^{2}+3\).

Answer:

Width \(=2x^{2}\), Length \(=7x^{2}+3\)