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. $3x^2 + 8x + 7$ (with a diagram of three colored rectangles labeled with $7x^2$ on the side and $3x^2$, $8x$, $7$ on top) area = blank box

Explanation:

Step1: Recall the area formula for a rectangle

The area \(A\) of a rectangle is given by \(A=\text{height}\times\text{width}\). Here, height \(h = 7x^{2}\) and width \(w=3x^{2}+8x + 7\). So, \(A=7x^{2}(3x^{2}+8x + 7)\).

Step2: Use the distributive property (\(a(b + c+d)=ab+ac + ad\))

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Step3: Apply the rule of exponents \(a^{m}\times a^{n}=a^{m + n}\) and multiply the coefficients

For \(7x^{2}\times3x^{2}\): Coefficient \(7\times3 = 21\) and \(x^{2}\times x^{2}=x^{2 + 2}=x^{4}\), so \(7x^{2}\times3x^{2}=21x^{4}\).
For \(7x^{2}\times8x\): Coefficient \(7\times8 = 56\) and \(x^{2}\times x=x^{2+1}=x^{3}\), so \(7x^{2}\times8x = 56x^{3}\).
For \(7x^{2}\times7\): Coefficient \(7\times7=49\) and \(x^{2}\times1=x^{2}\), so \(7x^{2}\times7 = 49x^{2}\).

Step4: Combine the terms

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

Answer:

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