Problem 78 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ (x-y)(2 x+y) $$

Step-by-Step Solution

Answer

Answer: $2x^2 - xy - y^2$

Step 1: Distribute the first term of the first binomial

To multiply \((x-y)(2x+y)\), first distribute the \(x\) from the first binomial across the second binomial: $$ x(2x) + x(y) $$

Step 2: Distribute the second term of the first binomial

Now, distribute the \(-y\) from the first binomial across the second binomial: $$ -y(2x) - y(y) $$

Step 3: Combine the results of step 1 and step 2

Put the results of step 1 and step 2 together: $$ x(2x) + x(y) - y(2x) - y(y) $$

Step 4: Perform the multiplications

Now, multiply each term: $$ 2x^2 + xy - 2xy - y^2 $$

Step 5: Combine like terms

Combine the terms \(+xy\) and \(-2xy\): $$ 2x^2 - xy - y^2 $$ The result is: $$ 2x^2 - xy - y^2 $$

Key Concepts

Binomial Multiplication

Binomial multiplication involves multiplying two expressions, each containing two terms. In the original exercise, we're dealing with \((x - y)(2x + y)\). The goal is to apply multiplication across each term in the binomials. This involves the FOIL method, a popular way to remember how to multiply:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

For example, in our problem, the first term multiplication is \(x \times 2x\), which gives \(2x^2\). The other multiplications, following the initial steps, complete the binomial multiplication process.

Distribution in Algebra

Distribution is a method used to eliminate parentheses by spreading each term across the terms of another expression. In contexts like our exercise, it involves distributing each term of the first binomial to every term of the second binomial.

This means in \((x - y)(2x + y)\), each term in the first pair, \(x\) and \(-y\), must be multiplied with each term of the second pair, \(2x\) and \(y\).

• First, \(x\) is multiplied with each term in the second binomial, yielding \(x(2x) + x(y)\).
• Then \(-y\) is distributed in the same way, resulting in \(-y(2x) - y(y)\).

Distribution breaks the multiplication into manageable parts.

Combining Like Terms

In algebra, combining like terms is crucial for simplifying expressions. Like terms are terms that contain the same variables raised to the same power. In our multiplication problem, after distribution and multiplication, we're left with the expression \(2x^2 + xy - 2xy - y^2\).

To combine like terms:

• Identify terms that have identical variable components, such as \(xy\) and \(-2xy\).
• Perform addition or subtraction on the coefficients of these terms. Here, \(+xy - 2xy\) simplifies to \(-xy\).
• Combine them to simplify the expression to get \(2x^2 - xy - y^2\).

This simplification process makes expressions more manageable and easier to interpret.