Problem 46 - Elementary Algebra

Question

For the following problems, rewrite each phrase using algebraic notation. \((x+b)\) times \((x+7)\)

Step-by-Step Solution

Answer

Question: Rewrite the given phrase "(\(x+b\)) times (\(x+7\))" using algebraic notation and simplify the result. Answer: \(x^2 + 7x + bx + 7b\)

Step 1: Rewrite the Phrase in Algebraic Notation

We are given the phrase "(\(x+b\)) times (\(x+7\))" and we need to rewrite this using algebraic notation. In algebra, "times" means to multiply, so we can rewrite this phrase as a multiplication expression: (\(x + b\))(\(x + 7\))

Step 2: Apply the FOIL Method

To multiply these two binomials, we will use the FOIL method. FOIL stands for First, Outer, Inner, and Last, which refers to the terms we should multiply in the binomials. First: Multiply the first terms of each binomial. \(x \cdot x = x^2\) Outer: Multiply the outer terms of each binomial. \(x \cdot 7 = 7x\) Inner: Multiply the inner terms of each binomial. \(b \cdot x = bx\) Last: Multiply the last terms of each binomial. \(b \cdot 7 = 7b\)

Step 3: Combine the Results

Next, we will add the results from the multiplication in the previous step: \(x^2 + 7x + bx + 7b\)

Step 4: Simplify the Expression

Since there are no like terms in the expression, it is already simplified. The algebraic notation for the given phrase is: \(x^2 + 7x + bx + 7b\)

Key Concepts

Binomials

In algebra, a binomial is a type of polynomial that contains exactly two terms separated by a plus or minus sign. The expression \(x + b\) is binomial because it contains two terms: \(x\) and \(b\). These terms are grouped together by an addition sign. Binomials serve as building blocks for more complex algebraic expressions and are quite common in algebraic operations.

Understanding binomials is fundamental because they often appear in various algebraic operations, including multiplication and factorization.

• Examples: \(x + 1\), \(2a - 3\), \(y^2 + 4y\)
• Usage: They are used in quadratic equations, polynomial identities, and even used to approximate values.

Recognizing binomials helps in identifying their use in algebraic expressions and applying appropriate mathematical operations like multiplying binomials which often involves methods like FOIL.

FOIL Method

The FOIL Method is a mnemonic for a specific technique used to multiply two binomials. It's an acronym that stands for First, Outer, Inner, Last. Each word corresponds to a specific part of the two binomials being multiplied.

When you apply the FOIL method, you follow these steps:

• First: Multiply the first terms in each binomial. For \((x + b)\) and \((x + 7)\), this is \(x \cdot x = x^2\).
• Outer: Multiply the outer terms in the product. Here, it's \(x \cdot 7 = 7x\).
• Inner: Multiply the inner terms. That is \(b \cdot x = bx\).
• Last: Multiply the last terms in each binomial. Which gives \(b \cdot 7 = 7b\).

Once all these products are calculated, you combine them to achieve the final product of the binomials: \(x^2 + 7x + bx + 7b\). The FOIL method provides a systematic way to ensure all components are multiplied correctly, resulting in accurate expansion.

Multiplication in Algebra

Multiplication in algebra extends beyond simple arithmetic to include variables, constants, and expressions like binomials. When multiplying algebraic expressions, especially polynomials like binomials, understanding the foundational principles is crucial.

Some key aspects of algebraic multiplication include:

• Distributive Property: This is essential for multiplying expressions. It states that \(a(b + c) = ab + ac\).
• Combining Like Terms: After multiplication, it's important to simplify expressions by combining terms that are similar, although in the current example, no like terms appear so the expression is already simplified.
• Variable Powers: When multiplying like bases, add their exponents, for example, \(x \cdot x = x^2\).

These principles help in solving algebraic problems and are particularly useful in tasks requiring the multiplication of complex algebraic expressions such as multiplying binomials using the FOIL method or other algebraic strategies.