Problem 16 - Algebra and Trigonometry

Question

State the property of real numbers being used. \((x+a)(x+b)=(x+a) x+(x+a) b\)

Step-by-Step Solution

Answer

Distributive Property.

Step 1: Identify the Property

The equation \((x+a)(x+b)=(x+a)x+(x+a)b\) shows the distribution of one factor over the sum of two other terms. This is a form of distribution of multiplication over addition.

Step 2: State the Property

The property used here is the Distributive Property. This property states that for all real numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). This allows us to multiply a single term by each individual term inside a set of parentheses.

Step 3: Apply the Property

Using the Distributive Property, \((x+a)(x+b)\) can be expanded by distributing \((x+a)\) to both \(x\) and \(b\), resulting in \((x+a)x + (x+a)b\).

Key Concepts

Real Numbers

Real numbers are an essential topic in mathematics, covering almost every number you can think of. They include:

• Natural numbers (1, 2, 3,...)
• Whole numbers (0, 1, 2, 3,...)
• Integers (...-3, -2, -1, 0, 1, 2, 3,...)
• Rational numbers (fractions or decimals that repeat)
• Irrational numbers (decimals that never repeat or end, like \(\pi\) or \(\sqrt{2}\))

Real numbers can be found on the number line, stretching from negative infinity to positive infinity. Understanding them is crucial as they are used in everyday counting, measurement, and even complex calculations. In the context of the Distributive Property, all numbers involved—such as \(x\), \(a\), and \(b\) in our exercise—are real numbers. This ensures that operations and properties like multiplication and distribution are well-defined and consistent without any surprises. Real numbers play a foundational role in all aspects of algebra.

Multiplication Over Addition

The idea of multiplication over addition is at the heart of the Distributive Property. This principle is what allows terms within parentheses to be multiplied individually by terms outside the parentheses. In our exercise, we see the equation:\[(x+a)(x+b) = (x+a)x + (x+a)b\].This equation emphasizes how each term in \((x+b)\) is separately multiplied by \((x+a)\). Here’s how it works step-by-step:

• First, multiply the term \((x+a)\) by the first term \(x\).
• Next, multiply \((x+a)\) by the second term \(b\).
• Add the two products together to get the expanded form.

This distribution ensures that all components interact in a straightforward manner, ensuring no part of the equation is left out. The distributive approach streamlines solving and simplifying problems where addition and multiplication interact.

Properties of Operations

Properties of operations govern how numbers interact and allow us to perform algebraic manipulations confidently. The main properties are:

• Commutative Property: Involves the order of numbers. For addition, \(a + b = b + a\), and for multiplication, \(ab = ba\).
• Associative Property: Involves the grouping of numbers. For addition, \((a + b) + c = a + (b + c)\), and for multiplication, \((ab)c = a(bc)\).
• Distributive Property: Connects multiplication and addition, as explained in the exercise, \(a(b+c) = ab + ac\).

The Distributive Property is particularly useful because it provides a method to "distribute" multiplication over a sum or difference, facilitating the logical progression of combining terms and simplifying expressions. Understanding these properties helps in organizing calculations to be easier and more efficient, whether tackling simple arithmetic or complex algebraic expressions.