Question

the rectangle below has an area of 8x^5 + 12x^3 + 20x^2. the width of the rectangle is equal to the greatest common monomial factor of 8x^5, 12x^3, and 20x^2. what is the length and width of the rectangle? width = length =

Explanation:

Step1: Find GCF of coefficients

Find GCF of 8, 12, 20. Factors of 8 are 1, 2, 4, 8; of 12 are 1, 2, 3, 4, 6, 12; of 20 are 1, 2, 4, 5, 10, 20. GCF is 4.

Step2: Find GCF of variables

For $x^5$, $x^3$, $x^2$, using rule of exponents, GCF of variables is $x^2$.

Step3: Find GCF of monomials

The GCF of $8x^5$, $12x^3$, $20x^2$ is $4x^2$, so width $w = 4x^2$.

Step4: Find length

Since area $A=l\times w$ and $A = 8x^5+12x^3 + 20x^2$, $w = 4x^2$, then $l=\frac{8x^5+12x^3 + 20x^2}{4x^2}$.
Dividing each term: $\frac{8x^5}{4x^2}+\frac{12x^3}{4x^2}+\frac{20x^2}{4x^2}=2x^3 + 3x+5$.

Answer:

Width = $4x^2$
Length = $2x^3 + 3x + 5$