Math Problem Statement

How do you set up the expression to find the volume of the rectangular prism?

Solution

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Question 1:

You are asked to write a polynomial in standard form to represent the volume of the rectangular prism. The dimensions of the prism are given as:

  • Length = x1x - 1
  • Width = 2x+32x + 3
  • Height = xx

Formula for the volume of a rectangular prism:

V=Length×Width×HeightV = \text{Length} \times \text{Width} \times \text{Height} Substituting the given expressions for length, width, and height, we get: V(x)=(x1)×(2x+3)×xV(x) = (x - 1) \times (2x + 3) \times x

Now, let's expand this expression step by step.

  1. First, multiply the two binomials (x1)(2x+3)(x - 1)(2x + 3): = 2x^2 + 3x - 2x - 3 = 2x^2 + x - 3$$
  2. Now, multiply the result by xx (the height): V(x)=(2x2+x3)×x=2x3+x23xV(x) = (2x^2 + x - 3) \times x = 2x^3 + x^2 - 3x

So, the polynomial in standard form representing the volume of the prism is: V(x)=2x3+x23xV(x) = 2x^3 + x^2 - 3x


Question 2:

You are asked to find the product of (3x2)(5x)(3 - x^2)(5 - x).

We use the distributive property (FOIL method) to expand the two binomials:

= 15 - 3x - 5x^2 + x^3$$ Simplifying the terms, we get: $$= x^3 - 5x^2 - 3x + 15$$ Thus, the product of $$(3 - x^2)(5 - x)$$ is: $$x^3 - 5x^2 - 3x + 15$$ --- Would you like further clarification on any step? Here are some related questions to explore: 1. How do you factor a cubic polynomial like $$2x^3 + x^2 - 3x$$? 2. What is the general method for finding the volume of other shapes? 3. Can the volume polynomial $$V(x) = 2x^3 + x^2 - 3x$$ have real roots? 4. How does polynomial multiplication differ from factoring? 5. How can you visualize the volume of a rectangular prism with non-integer dimensions? **Tip:** Always double-check your polynomial expansions by reversing the steps (factoring or using FOIL in reverse) to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Expansion
Geometry

Formulas

Volume of Rectangular Prism: V = Length × Width × Height

Theorems

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Suitable Grade Level

Grades 8-10