Problem 12 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The distributive property is used.

Step 1: Identify the Operation

The given equation involves multiplication on both sides. On the left, two binomials are being multiplied, and on the right, a distributive property appears to be used.

Step 2: Recognize the Property

The transformation from \( (x+a)(x+b) \)to \( (x+a)x + (x+a)b \)is an application of the distributive property of real numbers, which states that for any real numbers \(a, b, c\), \(a(b+c) = ab + ac\). In this case:- \((x+a)\) is distributed over \((x+b)\) to yield \((x+a)x + (x+a)b\).

Step 3: Verification

To ensure the correctness of recognizing the property, we can verify if distributing \((x+a)\) across \(x\) and \(b\) as seen in the equation matches the distributive property: \[ (x+a)(x+b) = (x+a)x + (x+a)b \]This verification shows consistency with the property, confirming our identification.

Key Concepts

Properties of Real Numbers

The properties of real numbers include a series of fundamental rules that apply to addition, subtraction, multiplication, and division. These rules help us understand and manipulate numbers in different mathematical expressions. One of the most essential properties is the distributive property. This property is particularly useful in algebra when rearranging and simplifying expressions. By knowing these properties, students can solve equations and perform calculations more efficiently.
Some key properties of real numbers are:

• Commutative Property: This property states that the order of addition or multiplication does not affect the result. For example, \( a + b = b + a \) and \( ab = ba \).
• Associative Property: The way numbers are grouped in addition or multiplication does not change the outcome. For instance, \( (a + b) + c = a + (b + c) \) and \( (ab)c = a(bc) \).
• Distributive Property: This property applies when multiplying a sum by a number, which can be distributed to each addend of the sum. For example, \( a(b + c) = ab + ac \).

Recognizing these properties allows mathematicians to solve complex problems by breaking them down into simpler steps.

Multiplication of Binomials

Multiplying binomials is an important skill in algebra, which involves expanding expressions of the form \((x+a)(x+b)\). This operation is crucial for simplifying and solving equations. When you multiply binomials, you essentially apply the distributive property to each term in the first binomial across the terms in the second binomial.
Here's how to multiply binomials step-by-step:

• First, distribute the first term of the first binomial across both terms of the second binomial. For example, in \((x+a)(x+b)\), distribute \(x\) over both \(x\) and \(b\).
• Next, distribute the second term of the first binomial across both terms of the second binomial. In our example, distribute \(a\) over \(x\) and \(b\).
• Finally, combine like terms, if necessary, to simplify the expression further.

By following these steps, you can handle multiplication of binomials efficiently and accurately, leading to solutions for larger algebraic problems.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols, but no equals sign. It forms the foundation of algebra by representing quantities and relationships using variables and operations.
Algebraic expressions are used to express real-world problems in a mathematical form, facilitating problem-solving across various fields. They allow us to:

• Represent unknown values, often using variables like \(x\), \(y\), or \(z\).
• Model situations mathematically to predict and analyze outcomes.
• Simplify complex problems through manipulation based on algebraic rules.

Algebraic expressions are fundamental in forming equations and inequalities, which are essential in mathematical reasoning and decision-making.

Step-by-Step Solution

Breaking down complex problems step-by-step is a valuable problem-solving approach in mathematics. It makes challenging concepts more approachable and manageable, particularly when working with algebraic expressions and operations like the multiplication of binomials.
The exercise given uses a step-by-step solution to analyze and solve the equation \((x+a)(x+b)=(x+a)x+(x+a)b\). Here's how the steps unfold:

• Step 1: Identify the Operation - First, recognize the multiplication of binomials on the left side of the equation and note the application of a property on the right side.
• Step 2: Recognize the Property - It is important to identify the distributive property as the key element being used. This involves understanding that a single term is being distributed across a sum inside the parentheses.
• Step 3: Verification - Finally, confirm the application by checking if it matches the rules of the distributive property. Ensure consistency in the transformation to validate the solution.

This structured approach ensures clarity and precision in solving algebraic problems, allowing for better understanding and confidence in handling equations.