Problem 49 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(2 x-5 y)(x+3 y)$$

Step-by-Step Solution

Answer

The product is \(2x^2 + xy - 15y^2\).

Step 1: Apply Distributive Property

To multiply the binomials \((2x - 5y)(x + 3y)\), we first use the distributive property, also known as the FOIL method (First, Outer, Inner, Last). This method involves multiplying each term in the first binomial by each term in the second binomial.

Step 2: Multiply First Terms

Multiply the first terms of each binomial: \(2x \times x = 2x^2\).

Step 3: Multiply Outer Terms

Multiply the outer terms of the binomials: \(2x \times 3y = 6xy\).

Step 4: Multiply Inner Terms

Multiply the inner terms of the binomials: \(-5y \times x = -5xy\).

Step 5: Multiply Last Terms

Multiply the last terms of each binomial: \(-5y \times 3y = -15y^2\).

Step 6: Combine Like Terms

Combine the like terms from the products obtained in previous steps: \(2x^2 + 6xy - 5xy - 15y^2\). The middle terms \(+6xy\) and \(-5xy\) combine to \(+1xy\): \[2x^2 + 1xy - 15y^2\].

Step 7: Write the Final Expression

The final expression after combining the like terms is \[2x^2 + xy - 15y^2\].

Key Concepts

Distributive Property

The Distributive Property is a fundamental principle in algebra that makes multiplying expressions more manageable. It states that to multiply a sum by a number, you multiply each addend by that number separately and then add or subtract the results. When applied to multiplying binomials, this property helps us to systematically distribute each term across another expression.

In the context of the exercise, the distributive property comes into play as we expand the expression oindent \((2x - 5y)(x + 3y)\).We do this by distributing each term in the first binomial across the terms in the second binomial.

This involves not just one multiplication but a series of them:

• First, multiply the first term from the first binomial by each term in the second binomial.
• Then, do the same with the second term of the first binomial.

Each multiplication step is crucial for ensuring that all interactions between the terms are accounted for, setting the stage for simplifying the expression down the line.

FOIL Method

The FOIL Method is a specific technique derived from the Distributive Property, tailor-made for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, and it serves as a mnemonic to remind us how many different products we must calculate.

Specifically, when we multiply oindent \((2x - 5y)(x + 3y)\),we follow these steps:

• First: Multiply the first terms from each binomial: \(2x \times x = 2x^2\).
• Outer: Multiply the outermost terms in the product: \(2x \times 3y = 6xy\).
• Inner: Multiply the innermost pair of terms: \(-5y \times x = -5xy\).
• Last: Finally, multiply the last terms in each binomial: \(-5y \times 3y = -15y^2\).

Following the FOIL method ensures that we capture all terms needed for the subsequent steps of simplifying and combining like terms.

Combining Like Terms

After expanding expressions using the Distributive Property or the FOIL method, you often end up with multiple terms. Combining like terms helps simplify the expression by grouping terms with the same variable and exponent. These like terms can be added or subtracted.

In our example \(2x^2 + 6xy - 5xy - 15y^2\), we see terms that share variables and exponents:

• Terms \(6xy\) and \(-5xy\) are like terms because they both contain \(xy\).

To combine \(6xy - 5xy\), simply add oindent\(6xy - 5xy = 1xy\),resulting in \(+xy\).

Consequently, our expression simplifies to:oindent \(2x^2 + xy - 15y^2\).Combining like terms is essential for reducing expressions to their simplest form and arriving at a neat and concise final expression. This step refines the expression, revealing fewer terms that are easier to interpret and use in further calculations.