Problem 17 - Elementary Algebra

Question

Find the following products and simplify. $$ (x+y)(2 x-3 y) $$

Step-by-Step Solution

Answer

Question: Multiply and simplify the following polynomials: $$(x+y)(2x-3y)$$ Answer: The simplified product of the given polynomials is $$2x^2 - 1xy - 3y^2$$.

Step 1: Distribute first terms

Multiply the first terms of each polynomial together. In this case, it's x * 2x which equals 2x^2.

Step 2: Distribute outer terms

Multiply the outer terms of each polynomial together. Here, it's x * (-3y) which results in -3xy.

Step 3: Distribute inner terms

Multiply the inner terms of each polynomial together. In this case, it's y * 2x which equals 2xy.

Step 4: Distribute last terms

Multiply the last terms of each polynomial together. Here, it's y * (-3y) which results in -3y^2.

Step 5: Combine like terms

Now, we need to combine like terms to simplify the expression. In this case, we have -3xy and 2xy. Combining the two results in -1xy.

Step 6: Write the final expression

Put the terms from step 1, 5, and 4 together to get the final expression: $$ 2x^2 - 1xy - 3y^2 $$ The simplified product of the given polynomials is $$ 2x^2 - 1xy - 3y^2 $$.

Key Concepts

Distributive Property

Understanding the distributive property is essential when multiplying polynomials. It allows us to multiply each term of one polynomial by every term of another polynomial. The classic representation of the distributive property is \( a(b + c) = ab + ac \). This means that when you have a single term outside of a parenthesis, you must multiply it by each term inside the parenthesis.

In the context of our exercise, when we have \( (x+y)(2x-3y) \), we distribute the terms as follows: The term \( x \) multiplies both \( 2x \) and \( -3y \) individually, and the term \( y \) also multiplies both \( 2x \) and \( -3y \) individually, creating four products. It's the equivalent of saying \( x \) shakes hands with \( 2x \) and then with \( -3y \) before \( y \) does the same. This careful distribution is what ensures that every possible product is accounted for in our solution.

Combining Like Terms

Once we've used the distributive property and multiplied the terms accordingly, we will often find ourselves with a collection of terms that can be simplified further. Combining like terms is the process of merging terms that have the same variables raised to the same power, essentially 'grouping' similar items together. For instance, if you have multiple terms containing \( xy \), such as \( 2xy \) and \( -3xy \) from our distributed multiplication, they're candidates for combination.

In our exercise, we identified \( -3xy \) and \( 2xy \) as like terms because they both contain the product of \( x \) and \( y \). Adding these together gives us \( -1xy \). This step is comparable to merging coins of the same denomination: just as you would combine quarters with quarters, you combine \( xy \) terms with \( xy \) terms. This ensures that our expression is as simple and concise as possible.

Simplifying Expressions

The final stage of polynomial multiplication is simplifying the expression. This involves not only combining like terms, as was previously discussed, but also arranging terms in a standard form, often from highest power to lowest power. Simplifying makes it much easier for anyone to understand and further work with the expression.

In our example, after combining like terms, we are left with \( 2x^2 \) which is already a simple term, \( -1xy \), and \( -3y^2 \). Our simplified expression, \( 2x^2 - xy - 3y^2 \), is written in descending order of degree, which is the standard convention for polynomial expressions. When done consistently, simplifying expressions not only improves clarity but also prepares students for more complicated algebraic operations, such as factoring or solving equations.