Question

a rectangle has a height of (7x^2) and a width of (3x^2 + 8x + 7). express the area of the entire rectangle. your answer should be a polynomial in standard form.
image of three colored rectangles with labels (3x^2), (8x), (7) and height (7x^2)
area =
box

Explanation:

Step1: Recall area formula for rectangle

The area \( A \) of a rectangle is given by the product of its height and width. So, \( A=\text{height}\times\text{width} \). Here, height is \( 7x^{2} \) and width is \( 3x^{2}+8x + 7 \). So we need to compute \( 7x^{2}(3x^{2}+8x + 7) \).

Step2: Apply distributive property (multiplication)

Using the distributive property \( a(b + c + d)=ab+ac + ad \), where \( a = 7x^{2} \), \( b=3x^{2} \), \( c = 8x \), and \( d=7 \).

• Multiply \( 7x^{2} \) with \( 3x^{2} \): \( 7x^{2}\times3x^{2}=21x^{4} \) (using the rule \( x^{m}\times x^{n}=x^{m + n} \), so \( x^{2}\times x^{2}=x^{4} \) and \( 7\times3 = 21 \))
• Multiply \( 7x^{2} \) with \( 8x \): \( 7x^{2}\times8x = 56x^{3} \) (using \( x^{2}\times x=x^{3} \) and \( 7\times8 = 56 \))
• Multiply \( 7x^{2} \) with \( 7 \): \( 7x^{2}\times7=49x^{2} \)

Step3: Combine the terms

Combine the results from the above multiplications: \( 21x^{4}+56x^{3}+49x^{2} \)

Answer:

\( 21x^{4}+56x^{3}+49x^{2} \)