Problem 13 - 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. $$(x+6)(x+10)$$

Step-by-Step Solution

Answer

The product is \(x^2 + 16x + 60\).

Step 1: Identify the Binomials

The problem involves multiplying two binomials: \((x + 6)\) and \((x + 10)\). We will use the distributive property (FOIL method) to find the product.

Step 2: Apply First Terms Multiplication

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

Step 3: Apply Outer Terms Multiplication

Multiply the outer terms: \(x \times 10 = 10x\).

Step 4: Apply Inner Terms Multiplication

Multiply the inner terms: \(6 \times x = 6x\).

Step 5: Apply Last Terms Multiplication

Multiply the last terms: \(6 \times 10 = 60\).

Step 6: Combine Like Terms

Now, combine the results from the FOIL method: \(x^2 + 10x + 6x + 60\). Combine the like terms \(10x\) and \(6x\) to get \(16x\).

Step 7: Write the Final Expression

The product of the binomials \((x + 6)(x + 10)\) is \(x^2 + 16x + 60\).

Key Concepts

Distributive Property

The distributive property is a valuable tool in algebra, particularly when multiplying expressions. In the context of binomials, the distributive property tells us that every term in the first binomial should be multiplied by every term in the second binomial.
Think of it as spreading one binomial across the other. In \((x+6)(x+10)\), you distribute and multiply throughout so that nothing gets left behind.

• Start with the first term in the first binomial: multiply it to each term in the second binomial.
• Next, take the second term in the first binomial and repeat the process.

This breaking down helps simplify complex products and sets the foundation for the FOIL method.

FOIL Method

The FOIL method is a mnemonic that stands for First, Outer, Inner, Last. It is specifically designed to make multiplying two binomials straightforward.
Here's how each part of FOIL breaks down:

• **First:** Multiply the first terms: \(x \times x = x^2\).
• **Outer:** Multiply the outer terms: \(x \times 10 = 10x\).
• **Inner:** Multiply the inner terms: \(6 \times x = 6x\).
• **Last:** Multiply the last terms: \(6 \times 10 = 60\).

Once each multiplication is complete, you combine all terms to find the simplified expression. This method helps ensure that no part of the binomial product is overlooked.

Like Terms

Like terms are terms within an algebraic expression that have the same variable raised to the same power. When we multiply binomials and use the FOIL method, we often result in like terms, especially when dealing with linear expressions.
In the expression \(x^2 + 10x + 6x + 60\), the terms \(10x\) and \(6x\) are like terms because they both contain the variable \(x\) to the same degree (first degree).
Combining like terms is simple:

• Add or subtract the coefficients of like terms while keeping the variable part the same.
• For example, \(10x + 6x\) combines to \(16x\).

This step is essential for simplifying expressions to their neatest form.

Binomials

A binomial is a polynomial with exactly two terms. Binomials are a fundamental component of algebraic expressions and are often seen when solving equations or multiplying expressions.
In the exercise \((x+6)(x+10)\), each bracket contains a binomial: two terms connected by either addition or subtraction.
When dealing with binomials, there are some key points to remember:

• Each term in a binomial can be a number, a variable, or the product of numbers and variables.
• In multiplication, they often follow specific patterns, like the one we've seen with the FOIL method.

Understanding binomials and their properties helps us navigate more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables linked by operation symbols like addition, subtraction, multiplication, and division. They form the building blocks of algebraic equations and functions.
An algebraic expression can consist of:

• Numbers (constants).
• Variables (like \(x, y, z\)) that hold values yet to be determined.
• Operations (+, −, ×, ÷).

In the provided problem, \((x+6)(x+10)\) is an algebraic expression that involves multiplying two simpler expressions (binomials).
Breaking down such expressions into steps simplifies the solving process, revealing the underlying mathematical structure.