Problem 59 - Elementary Algebra Within Reach

Question

In Exercises 59?62, factor the expression. $$ x(3-x)-2(3-x) $$

Step-by-Step Solution

Answer

The factored form of the expression \(x(3-x)-2(3-x)\) is \((x -2)(3-x)\).

Step 1: Identify common binomial

The first step is to identify a common binomial in the given expression. We observe that the binomial \(3-x\) appears twice, once multiplied with \(x\) and next with \(-2\). Hence, \(3-x\) can be said to be a common factor.

Step 2: Apply distributive property

The distributive property of multiplication over addition helps us to rewrite and simplify the expression. We write \(x(3-x)\) as \(x * (3-x)\) and \(-2(3-x)\) as \(-2 * (3-x)\). Now, since \(3-x\) is common, we factor it out and we will get \((x -2)(3-x)\) as the factored expression.

Key Concepts

Distributive Property

The distributive property is a key concept often used to simplify algebraic expressions. It states that when you have a number or variable outside the parentheses, it can be multiplied by each term inside the parentheses, separately. In other words, if you have an expression like \( a(b+c) \), you can distribute \( a \) over \( b+c \), giving you \( ab + ac \).
This property is crucial when factoring expressions because it allows us to combine terms and recognize patterns. In the expression \( x(3-x) - 2(3-x) \), the distributive property lets us factor \((3-x)\) as a common term from both \( x(3-x) \) and \(-2(3-x)\), leading to the simplified form

• Notice the repetition of \( (3-x) \) in the expression
• Factor out the common binomial: \((x-2)(3-x)\)

By understanding and applying the distributive property, we can easily manipulate and simplify complex expressions, making them more straightforward to handle in algebraic operations.

Binomial Factor

A binomial refers to an algebraic expression that contains exactly two terms, like \(a + b\). Identifying binomial factors is essential for simplifying expressions and solving equations. By recognizing these pairs, students can apply factoring techniques effectively. In the given exercise, \(3-x\) is the binomial factor that appears in both parts of the expression.
Factoring an expression by locating a common binomial involves:

• Identifying repeating binomial components in the expression
• Factoring the binomial out of each term where it appears
• Consolidating the expression into a factored form

By factoring out the binomial \(3-x\), the expression \(x(3-x) - 2(3-x)\) reduces to \((x-2)(3-x)\). This lays the groundwork for solving more complex algebraic problems, making binomials an essential part of any math student's toolkit.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition and multiplication. Understanding how to manipulate these expressions forms the core of learning algebra. They are the building blocks for more complicated equations and functions.
Algebraic expressions can vary from simple, like \( x + 3 \), to complex combinations such as \( x(3-x) - 2(3-x) \). Factors, terms, coefficients, and variables are all parts of algebraic expressions.

• Factors: Elements multiplied together to form the expression
• Terms: Individual parts of the expression separated by addition or subtraction
• Coefficients: Numerical values multiplying a term
• Variables: Symbols representing numbers

In the context of the exercise, understanding that the expression consists of terms that can be factored greatly simplifies the problem. Recognizing patterns, like common factors within terms, allows us to break down the expression into simpler, more manageable pieces. Factoring then becomes a tool for reducing complicated expressions into their simpler, more elegant forms.