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Chapter 10: Problem 444

Factor the greatest common factor from each polynomial. $$3 a-24$$

Short Answer

Expert verified
3(a - 8)

Step by step solution

01

Identify the terms in the polynomial

The given polynomial is 3a - 24. There are two terms in this polynomial: 3a and -24.
02

Determine the greatest common factor (GCF)

To find the greatest common factor, list the factors of each term: - Factors of 3a: 3, a - Factors of -24: 1, 2, 3, 4, 6, 8, 12, 24, -1, -2, -3, -4, -6, -8, -12, -24. The common factors are 1 and 3. The greatest common factor is 3.
03

Factor the GCF out of each term

Divide each term by the greatest common factor (3): - For 3a: 3a 梅 3 = a - For -24: -24 梅 3 = -8 The polynomial can now be written as: 3(a) 3 -8)]
04

Combine the factored terms

Combine the GCF and the remaining factored polynomial, which gives the factored form: 3(a - 8)

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is an essential part of algebra that involves breaking down a polynomial into simpler factors. This process can help simplify equations and solve for variables more easily.

When we factor a polynomial, we are looking for common factors in each term. For example, in the polynomial given, we would identify the terms first: 3a and -24. Both terms share the common factor of 3.

Notice how we factor it out by dividing each term by 3. This makes our polynomial go from 3a - 24 to 3(a - 8). When we鈥檝e finished factoring, we鈥檝e converted it into a product of the common factor and a simpler polynomial.
Prealgebra Basics
Understanding prealgebra basics is crucial because they form the foundation of more advanced math topics. In prealgebra, we deal with fundamental arithmetic operations and understand various forms of numbers.

Here are a few key concepts:
  • Arithmetic Operations: Addition, subtraction, multiplication, and division.
  • Factors: Numbers that divide another number without leaving a remainder.
  • Greatest Common Factor (GCF): The largest factor that two or more numbers have in common.
For our problem, we used these basics to find the GCF of the terms 3a and -24. By listing the factors of each term and identifying the highest one they share, we could proceed to factor the polynomial effectively.
Polynomial Terms
Polynomial terms are the parts of a polynomial separated by addition (+) or subtraction (鈭�) signs. Each term consists of a coefficient (a number) and one or more variables (letters).

For example, in the polynomial 3a - 24:
  • 3a consists of the coefficient 3 and the variable a.
  • -24 is a constant term, which means it has no variable.
Terms in a polynomial can vary; some may be simple like 3a, while others can be more complicated like 5x^2y.

In our exercise, understanding the terms helped us identify the common factor. By breaking down the polynomial into 3a and -24, we could determine that 3 was the common factor and factored the polynomial accordingly.

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Most popular questions from this chapter

Simplify. $$\frac{2^{8}}{2^{2}}$$

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