Answered

Vanessa uses the expressions $\left(3 x^2+5 x+10\right)$ and $\left(x^2-3 x-1\right)$ to represent the length and width of her patio. Which expression represents the area ($lw$) of Vanessa's patio?


A$3 x^4-4 x^3-8 x^2-35 x-10$
B$3 x^4+14 x^3+22 x^2+25 x-10$
C$3 x^4-4 x^3+22 x^2+35 x+10$
D$3 x^4+14 x^3+28 x^2+35 x+10$

Verified Answer :

  • Identify the area as the product of length and width: \(A(x) = (3x^2 + 5x + 10)(x^2 - 3x - 1)\).
  • Apply the distributive property to multiply every term of the first expression by the second.
  • Combine like terms: \((3x^4) + (-9x^3 + 5x^3) + (-3x^2 - 15x^2 + 10x^2) + (-5x - 30x) + (-10)\).
  • Conclude that the resulting area expression is \(\boxed{3x^4 - 4x^3 - 8x^2 - 35x - 10}\).

Explanation

  1. Analyzing the Problem

Vanessa's patio is defined by length \(L(x) = 3x^2 + 5x + 10\) and width \(W(x) = x^2 - 3x - 1\). To find the area of a rectangle, we multiply its length and width: \(A(x) = L(x) \cdot W(x)\). Our objective is to calculate the product \((3x^2 + 5x + 10)(x^2 - 3x - 1)\) and identify the correct expanded polynomial from the given options.

  1. Distributive Multiplication

We expand the expression \((3x^2 + 5x + 10)(x^2 - 3x - 1)\) by applying the distributive property. This means we multiply every term in the first polynomial by every term in the second:

  1. Multiply \(3x^2\) by each term:

\(3x^2 \cdot x^2 = 3x^4\)
\(3x^2 \cdot (-3x) = -9x^3\)
\(3x^2 \cdot (-1) = -3x^2\)

  1. Multiply \(5x\) by each term:

\(5x \cdot x^2 = 5x^3\)
\(5x \cdot (-3x) = -15x^2\)
\(5x \cdot (-1) = -5x\)

  1. Multiply \(10\) by each term:

\(10 \cdot x^2 = 10x^2\)
\(10 \cdot (-3x) = -30x\)
\(10 \cdot (-1) = -10\)

  1. Combining Like Terms

Now, we aggregate all the products and combine like terms:

\[A(x) = 3x^4 - 9x^3 - 3x^2 + 5x^3 - 15x^2 - 5x + 10x^2 - 30x - 10\]

Grouping by powers of \(x\):

  • \(x^4\): \(3x^4\)
  • \(x^3\): \(-9x^3 + 5x^3 = -4x^3\)
  • \(x^2\): \(-3x^2 - 15x^2 + 10x^2 = -8x^2\)
  • \(x\): \(-5x - 30x = -35x\)
  • Constants: \(-10\)

Combining these gives the final polynomial: \(3x^4 - 4x^3 - 8x^2 - 35x - 10\).

  1. Selecting the Final Answer

Comparing our result to the given options, we find it matches the first choice exactly. Therefore, the area of Vanessa's patio is represented by the expression \(\boxed{3x^4 - 4x^3 - 8x^2 - 35x - 10}\).

Examples

In architectural design and landscaping, finding the area of non-standard spaces often involves multiplying variable-based expressions representing dimensions. For example, if a homeowner is designing a garden plot where length and width depend on a parameter \(x\) (like the size of a foundational brick or a tiling unit), using polynomial multiplication helps determine the total square footage. This allows for accurate estimation of materials needed, such as soil or pavers, as the design scales.

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