Problem 67 - College Algebra

Question

Find each product. $$ (x+5 y)(7 x+3 y) $$

Step-by-Step Solution

Answer

The product of \((x+5y)\) and \((7x+3y)\) is \(7x^2 + 38xy + 15y^2\).

Step 1: Apply FOIL Method - First

Multiply the first terms in each binomial. So, \(x \times 7x = 7x^2\)

Step 2: Apply FOIL Method - Outside

Multiply the outside terms. So, \(x \times 3y = 3xy\)

Step 3: Apply FOIL Method - Inside

Multiply the inside terms, which gives \(5y \times 7x = 35xy\)

Step 4: Apply FOIL Method - Last

Multiply the last terms in each binomial. So, \(5y \times 3y = 15y^2\)

Step 5: Combine Like Terms

Add all of the resulting terms together and combine like terms, which gives \(7x^2 + 38xy + 15y^2\)

Key Concepts

Binomial Multiplication

When dealing with binomial multiplication, especially in algebra, one efficient method is the FOIL method. This stands for First, Outside, Inside, and Last. It's a systematic way to multiply two binomials, like $(x+5y)(7x+3y)$.

• First: Multiply the first terms of each binomial, giving us $7x^2$ from $x$ and $7x$.
• Outside: Then multiply the outer terms: $x$ and $3y$ to get $3xy$.
• Inside: The inside terms are $5y$ and $7x$, which multiply to $35xy$.
• Last: Finally, multiply the last terms in each binomial, $5y$ and $3y$, resulting in $15y^2$.

Each of these steps gives you a part of the final answer, and together they provide a clear path to solving the problem. This structured approach can help simplify binomial multiplication.

Combine Like Terms

The final step in binomial multiplication often involves combining like terms to simplify the expression. Like terms are terms that have identical variables raised to the same power. In our example, the terms generated by the FOIL method were:

- $7x^2$ - $3xy$ - $35xy$ - $15y^2$
To combine like terms effectively, focus on terms with the **same variables**:

- Add $3xy$ and $35xy$ because they both contain $xy$. This results in $38xy$.

Now, the expression $7x^2 + 3xy + 35xy + 15y^2$ becomes $7x^2 + 38xy + 15y^2$.
Combining like terms is essential in simplifying algebraic expressions and making them easier to understand.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are used to represent many different mathematical situations. An expression like $(x + 5y)(7x + 3y)$ explains a specific relationship.

• **Variables:** Letters like $x$ and $y$ stand for unknown values.
• **Coefficients:** Numbers like $7$, $5$, and $3$ are constants that multiply the variables.
• **Terms:** Parts of the expression separated by plus or minus signs.

When multiplying binomials, creating intermediate terms (like $3xy$ or $7x^2$) and then simplifying to form a final expression helps you work with algebraic expressions more effectively.
Understanding each component of an algebraic expression can make solving more complex problems more manageable. With practice, navigating and manipulating these can become second nature.