Problem 57 - Elementary Algebra

Question

Simplify. $$ 6 x 2 y-3 x y 2+2 x 2 y-5 x y 2 $$

Step-by-Step Solution

Answer

The simplified expression is \(8x^2y - 8xy^2\).

Step 1: Identify and Separate Like Terms

We start by identifying like terms. The expression has two types of terms: those involving \(x^2y\) and those involving \(xy^2\). We can rewrite the expression as follows to separate them:\[ (6x^2y + 2x^2y) + (-3xy^2 - 5xy^2) \]

Step 2: Combine Like Terms for \(x^2y\)

Focus on the terms involving \(x^2y\):\[ 6x^2y + 2x^2y = (6+2)x^2y = 8x^2y \]

Step 3: Combine Like Terms for \(xy^2\)

Now, look at the terms involving \(xy^2\):\[ -3xy^2 - 5xy^2 = (-3-5)xy^2 = -8xy^2 \]

Step 4: Write the Simplified Expression

Combine the results from Steps 2 and 3 to write the final simplified expression:\[ 8x^2y - 8xy^2 \]

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the first and most important steps is combining like terms. Like terms are terms within an algebraic expression that have the same variable parts raised to the same powers. This means that although the coefficients of these terms might differ, their variable components do not.

To combine like terms, simply add or subtract their coefficients, keeping the variable part the same.

• For instance, in the expression \(6x^2y + 2x^2y\), both terms have the same variable part \(x^2y\).
• Thus, we can combine them by adding their coefficients: \(6 + 2 = 8\), giving us \(8x^2y\).

Similarly, for the terms \(-3xy^2\) and \(-5xy^2\):

• The variable part \(xy^2\) is identical in both terms.
• So, we combine them: \(-3 - 5 = -8\), resulting in \(-8xy^2\).

By using combining like terms, expressions become simplified, reducing confusion and making further calculations easier.

Polynomials

Polynomials are algebraic expressions that include variables raised to whole number powers and coefficients. They can contain constants, variables, and exponents that are non-negative integers.

Polynomials are classified based on the number of terms they have:

• A monomial has one term, like \(7x^2\).
• A binomial has two terms, like \(3x + 4\).
• A trinomial has three terms, such as \(x^2 - 5x + 6\).

The expression \(6x^2y - 3xy^2 + 2x^2y - 5xy^2\) is a polynomial with four terms.
Combining these terms by adding or subtracting like terms helps to simplify it to \(8x^2y - 8xy^2\), which is also a polynomial consisting of two terms.

Simplifying polynomials is crucial because it makes polynomial functions easier to understand and solve.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition or subtraction). Unlike an equation, it does not have an equals sign and thus does not express a relationship between two values.

Understanding and manipulating algebraic expressions involves recognizing patterns and applying operations systematically:

• Identify terms: The expression \(6x^2y - 3xy^2 + 2x^2y - 5xy^2\) is made up of four separate terms.
• Determine like terms: Terms with the same variable and power, such as \(x^2y\) and \(xy^2\).
• Efficiently combine these like terms to simplify the expression as a whole.

Through simplification, complex algebraic expressions can be reduced to simpler forms, like converting \(6x^2y - 3xy^2 + 2x^2y - 5xy^2\) to \(8x^2y - 8xy^2\).

This process helps students better analyze and solve mathematical problems by focusing on relevant parts of the expression.