In order to evaluate Elijah's process and results when multiplying the monomial \(-\frac{3}{4}x^2y\) by the binomial \(\frac{2}{5}x - \frac{8}{3}\), we need to analyze the correct steps of multiplication and identify any potential mistakes. Here’s a structured note about what Elijah did correctly and what errors he may have made:
Note on Elijah's Multiplication Process
What Elijah Did Correctly:
- Identified the Components: Elijah recognized that he was multiplying a monomial by a binomial. This is the correct setup for this type of problem.
- Used the Distributive Property: Elijah correctly applied the distributive property (also known as the FOIL method for binomials), which involves multiplying the single term from the monomial by each term in the binomial separately.
What Errors Elijah Hired:
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Multiplying the First Term Incorrectly:
- When multiplying \(-\frac{3}{4}x^2y\) by \(\frac{2}{5}x\), the expected calculation should be: \[ -\frac{3}{4}x^2y \cdot \frac{2}{5}x = -\frac{3 \cdot 2}{4 \cdot 5} x^{2+1}y = -\frac{6}{20} x^3y = -\frac{3}{10} x^3y \] If Elijah made a mistake here, the first part of his product would be incorrect.
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Multiplying the Second Term Incorrectly:
- The multiplication of \(-\frac{3}{4}x^2y\) with \(-\frac{8}{3}\) should yield: \[ -\frac{3}{4}x^2y \cdot -\frac{8}{3} = \frac{3 \cdot 8}{4 \cdot 3} x^2y = \frac{24}{12} x^2y = 2x^2y \] Any errors in this multiplication will also affect the final result.
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Combining Terms:
- After obtaining the results from multiplying both terms, Elijah would need to correctly combine them: \[ -\frac{3}{10}x^3y + 2x^2y \] If he had errors in either multiplication, the combination would also be faulty.
Conclusion: Elijah needs to carefully review both multiplication steps to ensure they are executed correctly. It's important that he pays close attention to signs (positive or negative) and the multiplication of fractions to arrive at an accurate final expression.
Encouragingly, Elijah is on the right track in using the distributive property, which is vital when multiplying different types of algebraic expressions. However, careful calculation and attention to detail will help him achieve the correct product.
This note highlights the correct approach while pointing out specific errors Elijah made in his calculations, allowing him to learn and improve.