Question

a rectangle has a height of $3b^{2}$ and a width of $4b^{4}+2b - 6$. express the area of the entire rectangle. your answer should be a polynomial in standard form. area =

Explanation:

Step1: Recall area formula

The area formula for a rectangle is $A = \text{height}\times\text{width}$. Here, height $h = 3b^{2}$ and width $w=4b^{4}+2b - 6$. So $A=3b^{2}(4b^{4}+2b - 6)$.

Step2: Apply distributive property

$A = 3b^{2}\times4b^{4}+3b^{2}\times2b-3b^{2}\times6$.

Step3: Use exponent - rule for multiplication

When multiplying terms with the same base $a^{m}\times a^{n}=a^{m + n}$. So $3b^{2}\times4b^{4}=12b^{2 + 4}=12b^{6}$, $3b^{2}\times2b = 6b^{2+1}=6b^{3}$, and $3b^{2}\times6 = 18b^{2}$.

Step4: Write the polynomial in standard form

$A=12b^{6}+6b^{3}-18b^{2}$.

Answer:

$12b^{6}+6b^{3}-18b^{2}$