Question

Can someone help me find the width and the length

Answer 1

Width of the rectangle = $5k^2$

Length of the rectangle = $3 k^{2}+7 k+4$

Solution:

Area of the rectangle = $15 k^{4}+35 k^{3}+20 k^{2}$

Width of the rectangle = greatest common factor of $15 k^{4}, 35 k^{3}, \text { and } 20 k^{2}$

Factor of $15k^4$ = $5\times3\times k^2 \times k^2$

Factor of $35k^3$ = $5\times7\times k^2 \times k$

Factor of $20k^2$ = $5\times4\times k^2$

Greatest common factor of $15 k^{4}, 35 k^{3}, \text { and } 20 k^{2}$

                       = $5\times k^2$

                       $=5k^2$

Width of the rectangle $=5k^2$

Area of the rectangle = length × width

$15 k^{4}+35 k^{3}+20 k^{2}=\text{length}\times 5k^2$

Divide by 5k² on  both sides.

$\frac{15 k^{4}+35 k^{3}+20 k^{2}}{5k^2} =\frac{\text{length}\times 5k^2}{5k^2}$

$\frac{5k^2(3 k^{2}+7 k+4 )}{5k^2} =\text{length}$

Cancel the common factor 5k², we get

$3 k^{2}+7 k+4 =\text{length}$

Switch the sides.

Length = $3 k^{2}+7 k+4$

Width of the rectangle = $5k^2$

Length of the rectangle = $3 k^{2}+7 k+4$