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Chapter 6: Problem 10

Factor the given expressions completely. $$3 a-3 b$$

Short Answer

Expert verified
The expression factors to \( 3(a - b) \).

Step by step solution

01

Identify Common Factors

Look at both terms in the expression: \( 3a \) and \( 3b \). Both terms have a common factor of \( 3 \).
02

Factor Out the Greatest Common Factor

Rewrite the expression by factoring out the greatest common factor, which is \( 3 \). This gives us \( 3(a - b) \).
03

Verify the Factored Expression

Multiply the factor back into the expression to verify correctness: \( 3(a-b) = 3a - 3b \), confirming the factorization is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
A common factor is a number or variable that evenly divides two or more terms in an algebraic expression. Identifying common factors is the first step in the factoring process. In simple terms, if each term in your expression shares the same number or variable, that number or variable is a common factor. For example, in the expression \(3a - 3b\), the terms "3a" and "3b" both contain the number 3 as a common factor.

Finding common factors can simplify expressions and is a foundational skill in algebra:
  • Break down each term to see the smaller parts.
  • Identify numbers or variables common to all terms.
  • Remember, there can be more than one common factor, but we usually look for the greatest one first.
By finding and utilizing common factors, you can simplify both simple and complex expressions effectively.
Greatest Common Factor
When factoring, identifying the greatest common factor (GCF) is crucial. The GCF of an algebraic expression is the largest factor that divides all of the terms without leaving a remainder. Being able to pinpoint the GCF helps in simplifying expressions to their most basic form. Taking our example, \(3a - 3b\), you can see the common factor is 3.

Finding the GCF involves:
  • Looking at the numerical coefficients and determining the largest number that divides them all.
  • Examining any shared variables in the terms and choosing the variable with the lowest exponent.
If two terms are 9x and 6, their GCF is 3, because 3 is the largest number that can divide both 9 and 6. Understanding and applying the GCF in algebra helps streamline solving equations and simplifies expressions, making solving them more approachable.
Algebraic Expressions
Algebraic expressions are combinations of constants, variables, and operations that together form a mathematical "phrase." They are the backbone of algebra, allowing us to generalize mathematical problems and solutions. In our example, the expression \(3a - 3b\) consists of two terms. Each term combines a constant and a variable.

Important elements to look at in algebraic expressions:
  • **Constants**: These are numbers on their own, representing fixed values.
  • **Variables**: Symbols (like a or b) representing numbers that can vary.
  • **Operators**: Such as addition (-), subtraction (+), multiplication (x), division (梅) which are used to combine terms.
Breaking down and understanding the components allows you to navigate and manipulate expressions more efficiently. Whether you're factoring or solving, a solid grasp of algebraic expressions is key to success in algebra.

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Most popular questions from this chapter

Solve for the indicated letter. Each of the given formulas arises in the technical or scientific area of study listed. \(P=\frac{R T}{V-b}-\frac{a}{V^{2}},\) for \(T \quad\) (thermodynamics)

Reduce each fraction to simplest form. Each is from the indicated area of application. \(\frac{m u^{2}-m v^{2}}{m u-m v} \quad\) (nuclear energy)

Find the required expressions. A person travels a distance \(d\) at an average speed \(v_{1}\), and then returns over the same route at an average speed \(v_{2} .\) Write an expression in simplest form for the average speed of the round trip.

Factor the expressions completely. In Exercises 73 and \(74,\) it is necessary to set up the proper expression. Each expression comes from the technical area indicated. \(p_{1} R^{2}-p_{1} r^{2}-p_{2} R^{2}+p_{2} r^{2} \quad\) (fluid flow)

Solve the given equations and check the results. $$\frac{2}{3}+\frac{3}{y}=\frac{5}{2 y}$$
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