Question

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

Explanation:

Step1: Recall area formula

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

Step2: Distribute $4x^{3}$

Using the distributive property $a(b + c + d)=ab+ac + ad$, we have $4x^{3}\cdot x^{3}+4x^{3}\cdot3x^{2}+4x^{3}\cdot2x$.

Step3: Apply exponent - rule

When multiplying terms with the same base $a^{m}\cdot a^{n}=a^{m + n}$. So $4x^{3}\cdot x^{3}=4x^{3 + 3}=4x^{6}$, $4x^{3}\cdot3x^{2}=(4\times3)x^{3+2}=12x^{5}$, and $4x^{3}\cdot2x=(4\times2)x^{3 + 1}=8x^{4}$.

Step4: Write in standard form

The polynomial in standard form is $4x^{6}+12x^{5}+8x^{4}$.

Answer:

$4x^{6}+12x^{5}+8x^{4}$