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Chapter 6: Problem 11

Factor the given expressions completely. $$5 a-5$$

Short Answer

Expert verified
The expression is factored as \(5(a - 1)\).

Step by step solution

01

Identify the Common Factor

Start by observing the terms in the expression. The expression is \(5a - 5\). Both terms share a common factor, which is 5.
02

Factor Out the Greatest Common Factor

Factor out the greatest common factor (GCF) from the expression. In this case, factor 5 out of both terms: \[ 5(a - 1) \] This is the factored form of the expression.

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Key Concepts

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

Common Factor
In algebra, a common factor is a value that divides each term of an expression without leaving a remainder. Recognizing common factors is crucial when simplifying expressions. For instance, in the expression \(5a - 5\), both terms have the number 5 as a common factor because 5 divides both 5 and itself neatly.

When we observe terms, looking for values or variables they share as common factors is the first step in simplifying them.
  • Identify numbers or variables that repeat in each term.
  • Check if there is a coefficient common to all terms.
By finding a common factor, you are laying the groundwork to simplify the expression further.
Greatest Common Factor
The greatest common factor (GCF) is the highest value that divides each term in an expression perfectly.

Factoring with the GCF can significantly simplify solving algebraic expressions and equations. For example, in the expression \(5a - 5\), the common factors of the terms are found, but 5 is the greatest that both 5 and 0 (since \(a\) represents any number) share. Therefore, we use 5 to simplify both terms simultaneously.

Here鈥檚 how you can determine the GCF:
  • List all factors of each term.
  • Identify the largest number common to all lists.
By factoring out the GCF, you streamline expressions, making them easier to manipulate or solve.
Expressions
An expression is a combination of numbers, variables, and operations (like addition or subtraction) that together form a mathematical phrase.

Expressions are the backbone of algebra, allowing representation and manipulation of real-world quantities with symbols. In expressions like \(5a - 5\), each part plays a role:
  • \(5a\) denotes 5 times a certain value \(a\).
  • The \(-5\) is a constant term that adjusts the expression's overall value.
Understanding each component's role within the expression helps comprehend the underlying mathematics.

Expressing complex problems with algebra helps us tackle them by using familiar operations and theorems.

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Most popular questions from this chapter

Factor the given expressions completely. $$28 x^{2}-700 y^{2}$$

Perform the indicated operations and simplify. $$\frac{2}{13}+\frac{6}{13}$$

Factor the given expressions completely. $$4 x^{5}+14 x^{3}-8 x$$

Factor the given expressions completely. $$x^{2}-5 x-6$$

After finding the simplest form of each fraction, explain why it cannot be simplified more. (a) \(\frac{2 x+3}{2 x+6}\) (b) \(\frac{2(x+6)}{2 x+6}\)
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