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Chapter 10: Problem 460

Factor the greatest common factor from each polynomial. $$24 q^{2}-12 q$$

Short Answer

Expert verified
The factored form is \( 12q(2q - 1) \).

Step by step solution

01

Identify the common factor

The first step is to identify the greatest common factor (GCF) in the polynomial. Look at the coefficients and the variable terms separately. The coefficients are 24 and 12, and their GCF is 12. The variable terms are both powers of q, and the GCF of these is the smaller power, which is q.
02

Factor out the GCF

Now that the GCF has been identified, factor it out from each term in the polynomial. Divide each term by the GCF and write the polynomial as the product of the GCF and the resulting polynomial. For the expression \( 24 q^{2}-12 q \), the GCF is 12q. Dividing each term: - \(24 q^{2} \div 12q = 2q\) - \(12 q \div 12q = 1\).Therefore, 24q^2 - 12q = 12q(2q - 1).

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Key Concepts

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

Greatest Common Factor (GCF)
Understanding the Greatest Common Factor (GCF) is essential in factoring polynomials.
The GCF is the largest number or expression that evenly divides all terms in a polynomial.
To find the GCF, we look at both the coefficients and the variables separately.

For instance, in the polynomial \(24q^2 - 12q\), the coefficients are 24 and 12. The GCF of 24 and 12 is 12.
For the variable terms \(q^2\) and \(q\), the GCF is the smaller power of \(q\), which is \(q\). Combining these, the GCF of the polynomial is 12q.
Recognizing the GCF allows us to factor out the common terms, simplifying the polynomial into a more manageable form.
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler terms that, when multiplied together, give back the original polynomial.
The most common method is to factor out the GCF.
Let's take the polynomial \(24q^2 - 12q\). We identified that the GCF is 12q.

To factor the polynomial, we divide each term by the GCF:
  • \(24q^2 \div 12q = 2q\)
  • \(12q \div 12q = 1\)
So, \(24q^2 - 12q = 12q(2q - 1)\).
By factoring out the GCF, we've simplified the polynomial from two terms into a product of a monomial and a binomial.
Prealgebra
Prealgebra lays the foundation for all future mathematics learning.
It introduces fundamental concepts like the GCF and basic polynomial operations.

In prealgebra, you'll learn to:
  • Determine the GCF of sets of numbers and expressions.
  • Understand variable expressions and their powers.
  • Apply operations to simplify expressions and solve equations.
When working with polynomials, prealgebra frequently involves tasks like identifying and factoring out the GCF, which shows how expressions can be made simpler.
This prepares students for more advanced topics like algebra, where these foundational skills are applied in more complex scenarios.

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