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Chapter 5: Problem 24

Factor, if possible. $$ 24 y-30 $$

Short Answer

Expert verified
The expression factors to \(6(4y - 5)\).

Step by step solution

01

Find the Greatest Common Factor (GCF)

Identify the greatest common factor of the terms in the expression. The numbers in the expression are 24 and 30. Determine the GCF of these numbers. The greatest common factor of 24 and 30 is 6.
02

Factor Out the GCF

Use the GCF to factor the expression. Divide both terms by the GCF and express the original expression as a product of the GCF and a parenthesis containing the simplified terms.\[24y - 30 = 6(4y - 5)\]

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Key Concepts

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

Greatest Common Factor
When working with algebraic expressions, one of the foundational concepts is the notion of the Greatest Common Factor (GCF). The GCF of a set of numbers is the largest number that divides each of the numbers without leaving a remainder. In the exercise given, we worked with the numbers 24 and 30.
To find the GCF:
  • List the factors of each number.
  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
The greatest number common to both lists is 6, making it the GCF. Recognizing the GCF plays a crucial role in simplifying or factoring expressions, as it helps us see the structure within the numbers, leading to more manageable equations or polynomials.
Intermediate Algebra
In intermediate algebra, students build upon basic algebraic skills and delve into more complex topics such as factoring, polynomials, and rational expressions. It serves as a bridge between basic algebra and more advanced topics.
Key areas of focus include:
  • Simplifying expressions by factoring.
  • Solving quadratic equations.
  • Working with inequalities and absolute values.
Understanding how to factor expressions, like in our exercise, is fundamental to mastering intermediate algebra. It's not just about finding the solution, but also about understanding the process and the 'why' behind the steps. This strengthens analytical and problem-solving skills that are essential in higher levels of math.
Polynomial Factoring
Factoring polynomials is a technique in algebra that allows us to express a polynomial as a product of simpler polynomials. It's a crucial skill because it simplifies expressions, making solving equations easier.
The basic steps include:
  • Identify and extract the GCF of the polynomial terms, as demonstrated in our example expression, 24y - 30.
  • Rewrite the polynomial as a product of the GCF and the simplified polynomial, e.g., \(6(4y - 5)\).
  • If further simplification is possible, continue to factor the polynomial completely.
For beginners, this process might seem challenging because it involves a certain level of foresight and manipulation. However, with practice, recognizing patterns and performing steps becomes intuitive. Polynomial factoring is not just about simplifying; it sets the foundation for solving equations and understanding polynomial functions in-depth.

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Most popular questions from this chapter

Factor each difference of cubes. See Example 8. $$ x^{3}-216 y^{6} $$

Factor each expression. Factor out any GCF first. See Example 5. $$ 3 x^{3}-243 x $$

Write each number in standard notation. See Example 3. $$ 3.5 \times 10^{-7} $$

Factor expression completely. If an expression is prime, so indicate. \(8 x^{3} y^{4}-27 y\)

Factor, if possible. $$ 21 x^{2}+7 x $$
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