Question

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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 GCF of coefficients

Find GCF of 14 and 6. Factors of 14 are 1, 2, 7, 14. Factors of 6 are 1, 2, 3, 6. GCF is 2.

Step2: Find GCF of variable - parts

For $x^4$ and $x^2$, using the rule $x^m$ and $x^n$ ($m\geq n$), GCF is $x^2$.

Step3: Determine the greatest common monomial factor

The greatest common monomial factor of $14x^4$ and $6x^2$ is $2x^2$, so the width of the rectangle is $2x^2$.

Step4: Calculate the length

We know that area $A = l\times w$. Given $A=14x^4 + 6x^2$ and $w = 2x^2$. Then $l=\frac{14x^4 + 6x^2}{2x^2}$. Using the distributive property of division $\frac{a + b}{c}=\frac{a}{c}+\frac{b}{c}$, we have $l=\frac{14x^4}{2x^2}+\frac{6x^2}{2x^2}=7x^2 + 3$.

Answer:

Width = $2x^2$
Length = $7x^2+3$