/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 \(\left(x^{3}-2 x^{2} y\right)-\... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 62

\(\left(x^{3}-2 x^{2} y\right)-\left(x y^{2}-3 y^{3}\right)-\left(x^{2} y-4 x y^{2}\right)\)

Short Answer

Expert verified
The simplified expression is: \[(x^3 - 3 x^2 y + 3xy^2 + 3y^3)\]

Step by step solution

01

Distribute the Negative Sign

Start by distributing the negative signs inside the parentheses: ewline ewline \[(x^3 - 2x^2 y) - (x y^2 - 3 y^3) - (x^2 y - 4 x y^2) \rightarrow (x^3 - 2x^2 y) - x y^2 + 3 y^3 - x^2 y + 4 x y^2.\]
02

Combine Like Terms

Combine the like terms in the expression: ewline The like terms are: ewline \[-2x^2 y - x^2 y: -2x^2 y - x^2 y = -3 x^2 y, \] ewline \[-xy^2 + 4xy^2: -xy^2 + 4xy^2 = 3xy^2.\]
03

Simplify the Expression

Combine the results from Step 2 and simplify: ewline ewline \[(x^3 - 3 x^2 y + 3xy^2 + 3y^3)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributing negative signs
Understanding how to distribute negative signs is crucial in algebra. When we distribute a negative sign, we change the sign of each term inside the parentheses. For example, given \(( - (a + b) )\), we first multiply -1 by each term: \(-(a + b) = -a - b \). In our original exercise, we had to distribute negative signs across three sets of parentheses.
This resulted in: \[ (x^3 - 2x^2 y) - (x y^2 - 3 y^3) - (x^2 y - 4 x y^2) = (x^3 - 2x^2 y) - x y^2 + 3 y^3 - x^2 y + 4 x y^2 \]. By careful distribution, each term's sign was correctly adjusted, simplifying the next steps.
combining like terms
Combining like terms means adding or subtracting terms with the same variables and powers. This step streamlines the expression. In our exercise, the like terms are: \(-2x^2 y\) and \(-x^2 y.\) Combining these, we get: \(-2x^2 y - x^2 y = -3 x^2 y.\) Similarly, for \(-xy^2\) and \(+4xy^2,\) we have: \(-xy^2 + 4xy^2 = 3xy^2.\) This leaves us with:
-3 x^2 y and 3xy^2.
simplifying expressions
Simplifying expressions involves performing all possible operations to present the expression in its simplest form. After distributing the negative signs and combining like terms in our example, we were left with the equation:
\((x^3 - 3 x^2 y + 3xy^2 + 3y^3))\). Each step - distributing signs, combining terms - made the expression easier to manage until no further simplification was possible. This process helps in further algebraic manipulations or solving equations.
intermediate algebra
Intermediate algebra builds on foundational concepts and prepares you for advanced topics. It involves steps like:
  • Distributing negative signs
  • Combining like terms
  • Simplifying expressions
This is exactly what we did in the original exercise. Each skill interconnects, making mastering these essential for success in algebra. Practice these techniques regularly, as they form the backbone for solving more complex problems.

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Most popular questions from this chapter

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Find each product. $$ (5 x-y)(x-4) $$

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For functions \(\quad f(x)=2 x-7\) \(g(x)=2 x+7,\) find (a) \((f \cdot g)(x)\) (b) \((f \cdot g)(-3)\)
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