key term - Factoring Out Common Factors

Definition

Factoring out common factors is a technique in algebra where the greatest common factor (GCF) of a polynomial expression is identified and extracted, resulting in a simplified expression. This process helps to break down complex polynomial expressions into more manageable components.

5 Must Know Facts For Your Next Test

1. Factoring out common factors is a crucial step in the process of factoring polynomials, as it simplifies the expression and makes it easier to identify additional factors.
2. The GCF of a polynomial expression is the largest factor that is common to all the terms in the expression.
3. Factoring out common factors can be done by dividing each term in the expression by the GCF, and then multiplying the resulting expression by the GCF.
4. Factoring out common factors often leads to the identification of other factors, such as binomial factors, which can further simplify the expression.
5. The distributive property is an important concept in factoring out common factors, as it allows for the rearrangement of terms to reveal the GCF.

Review Questions

• Explain the process of factoring out common factors from a polynomial expression.
  • To factor out common factors from a polynomial expression, you first need to identify the greatest common factor (GCF) of all the terms in the expression. This involves finding the largest positive integer that divides each of the coefficients without a remainder. Once the GCF is identified, you can divide each term in the expression by the GCF, and then multiply the resulting expression by the GCF. This simplifies the expression and makes it easier to identify additional factors, such as binomial factors.
• Describe how the distributive property is used in the process of factoring out common factors.
  • The distributive property is an important concept in factoring out common factors. It states that multiplying a sum by a number is the same as multiplying each addend by that number and then adding the results. This property allows for the rearrangement of terms in a polynomial expression to reveal the GCF. By applying the distributive property, you can isolate the common factor and factor it out, leaving behind a simplified expression that can be further factored if necessary.
• Analyze the significance of factoring out common factors in the context of polynomial factorization.
  • Factoring out common factors is a crucial step in the process of factoring polynomials because it simplifies the expression and makes it easier to identify additional factors. By extracting the GCF, you can break down the polynomial into more manageable components, which can then be further factored using techniques like finding binomial factors. This process of breaking down complex polynomial expressions into simpler forms is essential for solving a wide range of algebraic problems and understanding the underlying structure of polynomial functions.

Related terms

Greatest Common Factor (GCF):

The largest positive integer that divides each of the coefficients in a polynomial expression without a remainder.

Polynomial Factorization: The process of expressing a polynomial as a product of simpler polynomial expressions, such as linear factors.

Distributive Property:

The mathematical property that states that multiplying a sum by a number is the same as multiplying each addend by that number and then adding the results.