Problem 43 - Beginning and Intermediate Algebra

Question

Use FOIL to multiply. $$(w+8)(w+7)$$

Step-by-Step Solution

Answer

Using the FOIL method to multiply \((w+8)(w+7)\), we obtain the simplified expression \(\boxed{w^2 + 15w + 56}\).

Step 1: Multiply the First terms

Multiply the first terms in each binomial: \(w\) from \((w+8)\) and \(w\) from \((w+7)\). Multiplication: \(w * w = w^2\)

Step 2: Multiply the Outer terms

Multiply the outer terms in each binomial: \(w\) from \((w+8)\) and \(7\) from \((w+7)\). Multiplication: \(w * 7 = 7w\)

Step 3: Multiply the Inner terms

Multiply the inner terms in each binomial: \(8\) from \((w+8)\) and \(w\) from \((w+7)\). Multiplication: \(8 * w = 8w\)

Step 4: Multiply the Last terms

Multiply the last terms in each binomial: \(8\) from \((w+8)\) and \(7\) from \((w+7)\). Multiplication: \(8 * 7 = 56\)

Step 5: Combine the results

Now, combine the results from Steps 1-4: \(w^2 + 7w + 8w + 56\)

Step 6: Simplify the expression

Add the like terms: \(w^2 + 15w + 56\) The resulting expression after using the FOIL method to multiply \((w+8)(w+7)\) is: \(\boxed{w^2 + 15w + 56}\)

Key Concepts

Binomial Multiplication

Understanding binomial multiplication is a crucial step in mastering algebra. A binomial is an algebraic expression that contains two terms, such as \((w+8)\).In binomial multiplication, where one binomial is multiplied by another, like \((w+8)(w+7)\), the goal is to find the product of these two expressions. The FOIL method, which stands for First, Outer, Inner, Last, is employed to systematically multiply these terms.

• The **First** step is multiplying the first terms of each binomial, \(w \cdot w\), resulting in \(w^2\).
• The **Outer** step involves the outer terms, \(w\) and \(7\), giving \(7w\).
• Next, the **Inner** step, focuses on \(8\) and \(w\), producing an \(8w\).
• The **Last** step multiplies the last terms, \(8\) and \(7\), to obtain \(56\).

For beginners, this systematic approach helps in organizing the multiplication process, ensuring no terms are missed. The steps are then combined to form a complete product of the binomials.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They are the building blocks of algebra, allowing us to generalize arithmetic operations. For instance, in the expression \((w+8)\), \(w\) is a variable that could represent any number, and \(8\) is a constant. In algebra, variables are used to represent unknown values and solve equations.

When dealing with expressions like \(w^2 + 15w + 56\), understanding the role of each component is essential.

• The term \(w^2\) is a product of the variable \(w\) multiplied by itself, highlighting the concept of powers.
• The coefficient \(15\) in \(15w\) indicates how many times the variable \(w\) is being used.
• The constant term \(56\) remains unchanged during variable manipulation.

Simplifying these expressions involves combining like terms, such as adding \(7w\) and \(8w\) to form \(15w\), refining the expression.

Polynomial Multiplication

Polynomial multiplication takes binomial multiplication a step further, involving expressions with more than two terms. When multiplying polynomials, each term of one polynomial is multiplied by each term of the other polynomial. This technique often builds on the binomial multiplication concept.

To understand polynomial multiplication, consider how the distributed property, \(a(b + c) = ab + ac\), is utilized but expanded to more terms.

• Start by identifying all terms involved in both polynomials.
• Next, systematically multiply each term in the first polynomial by every term in the second polynomial, ensuring nothing is left out.
• Finally, add the like terms resulting from the multiplication to simplify.

In our equation from the exercise, we exemplified polynomial multiplication with just two terms per binomial, resulting in a quadratic polynomial, which is a polynomial of degree 2. Mastery of polynomial multiplication is vital because it forms the foundation for more advanced algebraic concepts, ranging from factoring to solving complex equations.