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Chapter 4: Problem 107

Simplify or solve as appropriate. $$3 x y(x+y)-2 x(x y-x)$$

Short Answer

Expert verified
The simplified form of \(3xy(x+y)-2x(xy-x)\) is \(2x^2 + x^2y + 3xy^2\).

Step by step solution

01

Apply the Distributive Property

The first step is to apply the distributive property to both terms in the expression. This gives us \(3x^2y + 3xy^2 - 2x^2y + 2x^2\).
02

Combine Like Terms

Next, we combine like terms, meaning we add or subtract the coefficients (the numbers in front of the variables) of the terms with the same variables and powers. This gives us \((3x^2y - 2x^2y) + 3xy^2 + 2x^2 = x^2y + 3xy^2 + 2x^2\).
03

Reorder The Terms

Finally, as per custom, we reorder the terms of the expression in descending powers of x. This results in \(2x^2 + x^2y + 3xy^2\).

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Key Concepts

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

Distributive Property
The distributive property is a foundational principle in algebra that helps you expand expressions and work with them more easily. It states that for any three numbers or variables, \(a(b + c) = ab + ac\). You distribute the first term among the terms inside the parenthesis.

In our exercise, we start with:
  • \(3xy(x + y)\) becomes \(3x^2y + 3xy^2\).
  • \(-2x(xy - x)\) becomes \(-2x^2y + 2x^2\).
By applying the distributive property, we've removed the parentheses and rewritten the expression in a form that allows us to move to the next step. Make sure to multiply each term inside the parentheses by the term outside to ensure accuracy.
Combining Like Terms
Combining like terms is a method to simplify expressions that have similar terms, which means terms that have the exact same variables raised to the same powers. The process involves adding or subtracting the coefficients of these similar terms.

In this exercise, once the distributive property has been applied, we end up with:
  • \(3x^2y - 2x^2y + 3xy^2 + 2x^2\).
Here, \(3x^2y\) and \(-2x^2y\) are like terms because they contain the same variables with the same powers. When we combine them, they become \(x^2y\). This process of adding/subtracting coefficients simplifies the expression further, making it easier to manage.
Polynomial Reordering
Reordering terms in a polynomial often involves arranging them in a particular sequence, usually in descending order based on the powers of one of the variables, typically \(x\). This not only makes the polynomial easier to read but also aligns with mathematical convention.

After applying the distributive property and combining like terms, we simplified the expression to:
  • \(x^2y + 3xy^2 + 2x^2\).
To follow standard practices, reordering the terms means rearranging them by the degree of \(x\), leading to:
  • \(2x^2 + x^2y + 3xy^2\)
This step does not change the expression's value but makes it easier to compare, evaluate, or further simplify in algebraic contexts.

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Most popular questions from this chapter

Simplify or solve as appropriate. $$(x-3)^{2}-(x+3)^{2}$$

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