Question

the rectangle below has an area of 6n^4 + 20n^3 + 14n^2. the width of the rectangle is equal to the greatest common monomial factor of 6n^4, 20n^3, and 14n^2. what is the length and width of the rectangle?

Explanation:

Step1: Find GCF of coefficients

Find GCF of 6, 20, 14. Prime - factorize: $6 = 2\times3$, $20=2^{2}\times5$, $14 = 2\times7$. GCF of 6, 20, 14 is 2.

Step2: Find GCF of variable parts

For $n^{4},n^{3},n^{2}$, using the rule of exponents for GCF of powers of the same base ($a^{m}$ and $a^{n}$ with $m\geq n$, GCF is $a^{n}$), GCF of $n^{4},n^{3},n^{2}$ is $n^{2}$.

Step3: Determine the width

The greatest common monomial factor of $6n^{4},20n^{3},14n^{2}$ is $2n^{2}$, so the width $w = 2n^{2}$.

Step4: Calculate the length

We know that the area of a rectangle $A=l\times w$. Given $A = 6n^{4}+20n^{3}+14n^{2}$ and $w = 2n^{2}$. Then $l=\frac{A}{w}=\frac{6n^{4}+20n^{3}+14n^{2}}{2n^{2}}$.
Using the distributive property of division over addition $\frac{a + b + c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$, we have $\frac{6n^{4}}{2n^{2}}+\frac{20n^{3}}{2n^{2}}+\frac{14n^{2}}{2n^{2}}$.
Applying the rule of exponents $\frac{a^{m}}{a^{n}}=a^{m - n}$, we get $3n^{2}+10n + 7$.

Answer:

Width: $2n^{2}$, Length: $3n^{2}+10n + 7$