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.

(x^3 + 3x^2 + 2x)
(4x^3)

area =

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Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by the product of its height and width, i.e., \( A=\text{height}\times\text{width} \). Here, the height is \( 4x^{3} \) and the width is \( x^{3}+3x^{2}+2x \), so we need to calculate \( 4x^{3}(x^{3}+3x^{2}+2x) \).

Step2: Apply the distributive property (multiplication of a monomial by a polynomial)

Using the distributive property \( a(b + c + d)=ab+ac + ad \), where \( a = 4x^{3} \), \( b=x^{3} \), \( c = 3x^{2} \), and \( d=2x \), we get:
\[

$$\begin{align*} 4x^{3}(x^{3})+4x^{3}(3x^{2})+4x^{3}(2x)&=4x^{3 + 3}+12x^{3+2}+8x^{3 + 1}\\ &=4x^{6}+12x^{5}+8x^{4} \end{align*}$$

\]
(We used the exponent rule \( a^{m}\times a^{n}=a^{m + n} \) for each term.)

Answer:

\( 4x^{6}+12x^{5}+8x^{4} \)