Problem 12 - Algebra for College Students

Question

Translate each verbal phrase into \(a\) mathematical expression using \(x\) as the variable. The product of 8 more than a number and 5 less than the number.

Step-by-Step Solution

Answer

(x + 8)(x - 5)

Step 1: Identify the expressions for the phrases

The phrase '8 more than a number' can be written mathematically as \(x + 8\). The phrase '5 less than the number' can be written as \(x - 5\).

Step 2: Express the product

Now, multiply these two expressions together since 'the product' means multiplication: \( (x + 8) (x - 5) \).

Key Concepts

algebraic expressions

Algebraic expressions are a way to represent mathematical phrases using variables, numbers, and operations. In algebra, variables like \( x \) stand in for unknown values. An expression can include operations like addition, subtraction, multiplication, and division. For example, \( x + 8 \) and \( x - 5 \) are algebraic expressions. Knowing how to translate a phrase into an algebraic expression is a valuable skill, especially when dealing with word problems. It allows you to write math problems in a form that is easier to solve.

verbal phrases in math

Verbal phrases in math are sentences or parts of sentences that describe mathematical operations and relationships using words. Knowing how to translate these phrases into algebraic expressions is key. For instance, '8 more than a number' means you take an unknown number (\( x \)) and add 8 to it. This gives you the expression \( x + 8 \). Similarly, '5 less than a number' means you subtract 5 from an unknown number, resulting in the expression \( x - 5 \). By accurately translating verbal phrases, you can set up mathematical problems correctly and solve them more effectively.

product of expressions

The product of expressions means you are multiplying two or more algebraic expressions. In our example, we have the expressions \( x + 8 \) and \( x - 5 \). To find their product, multiply them together: \((x + 8)(x - 5)\). When you multiply these, you are using the distributive property—each term in the first expression must be multiplied by each term in the second. So:
\[ (x + 8)(x - 5) = x \times x + x \times (-5) + 8 \times x + 8 \times (-5) \]
Simplifying this gives:
\[ x^2 - 5x + 8x - 40 = x^2 + 3x - 40 \]
Understanding how to find the product of expressions is essential for solving many algebra problems, especially in polynomial arithmetic.