Math Problem Statement
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 =
- Width =
- Height =
Formula for the volume of a rectangular prism:
Substituting the given expressions for length, width, and height, we get:
Now, let's expand this expression step by step.
- First, multiply the two binomials : = 2x^2 + 3x - 2x - 3 = 2x^2 + x - 3$$
- Now, multiply the result by (the height):
So, the polynomial in standard form representing the volume of the prism is:
Question 2:
You are asked to find the product of .
We use the distributive property (FOIL method) to expand the two binomials:
= 15 - 3x - 5x^2 + x^3$$ 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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Expansion
Geometry
Formulas
Volume of Rectangular Prism: V = Length × Width × Height
Theorems
-
Suitable Grade Level
Grades 8-10
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