Width of the rectangle = [tex]5k^2[/tex]
Length of the rectangle = [tex]3 k^{2}+7 k+4[/tex]
Solution:
Area of the rectangle = [tex]15 k^{4}+35 k^{3}+20 k^{2}[/tex]
Width of the rectangle = greatest common factor of [tex]15 k^{4}, 35 k^{3}, \text { and } 20 k^{2}[/tex]
Factor of [tex]15k^4[/tex] = [tex]5\times3\times k^2 \times k^2[/tex]
Factor of [tex]35k^3[/tex] = [tex]5\times7\times k^2 \times k[/tex]
Factor of [tex]20k^2[/tex] = [tex]5\times4\times k^2[/tex]
Greatest common factor of [tex]15 k^{4}, 35 k^{3}, \text { and } 20 k^{2}[/tex]
= [tex]5\times k^2[/tex]
[tex]=5k^2[/tex]
Width of the rectangle [tex]=5k^2[/tex]
Area of the rectangle = length × width
[tex]15 k^{4}+35 k^{3}+20 k^{2}=\text{length}\times 5k^2[/tex]
Divide by 5k² on both sides.
[tex]$\frac{15 k^{4}+35 k^{3}+20 k^{2}}{5k^2} =\frac{\text{length}\times 5k^2}{5k^2}[/tex]
[tex]$\frac{5k^2(3 k^{2}+7 k+4 )}{5k^2} =\text{length}[/tex]
Cancel the common factor 5k², we get
[tex]3 k^{2}+7 k+4 =\text{length}[/tex]
Switch the sides.
Length = [tex]3 k^{2}+7 k+4[/tex]
Width of the rectangle = [tex]5k^2[/tex]
Length of the rectangle = [tex]3 k^{2}+7 k+4[/tex]
