Factor by Grouping

Definition

Factor by grouping is a technique used to factor polynomials by identifying common factors among groups of terms and then factoring out those common factors. This method is particularly useful when dealing with polynomials that have more than two terms and where the traditional method of factoring by finding the greatest common factor (GCF) may not be sufficient.

5 Must Know Facts For Your Next Test

1. Factor by grouping is a technique used to factor polynomials with more than two terms when the traditional method of finding the GCF is not sufficient.
2. The process of factor by grouping involves dividing the polynomial into groups of terms, finding the GCF of each group, and then factoring out the common factor.
3. Factor by grouping is particularly useful when dealing with polynomials that have a mix of positive and negative terms, or when the coefficients of the terms are not easily identifiable.
4. The technique of factor by grouping can be applied to both linear and quadratic polynomials, as well as higher-degree polynomials.
5. Factoring by grouping is an important skill in solving quadratic equations, as it can help simplify the expression and make it easier to find the roots or solutions.

Review Questions

• Explain the steps involved in factoring a polynomial by grouping.
  • To factor a polynomial by grouping, follow these steps: 1. Divide the polynomial into groups of terms, typically based on the common factors among the terms. 2. Find the greatest common factor (GCF) of each group of terms. 3. Factor out the GCF from each group, leaving behind a smaller polynomial in each group. 4. Combine the factored groups using the distributive property to obtain the final factored form of the original polynomial.
• How does the factor by grouping method differ from the traditional method of finding the greatest common factor (GCF)?
  • The factor by grouping method is used when the traditional GCF method is not sufficient, such as when dealing with polynomials with more than two terms or with a mix of positive and negative terms. While the GCF method focuses on finding the largest common factor among all the terms, the factor by grouping method involves dividing the polynomial into smaller groups, finding the GCF of each group, and then factoring out those common factors. This approach allows for more complex polynomials to be factored effectively.
• Explain how the factor by grouping technique can be applied to solve quadratic equations.
  • Factoring quadratic equations is a crucial step in solving them, and the factor by grouping method can be particularly useful in this context. By factoring the quadratic expression into a product of smaller polynomials, the equation can be solved by setting each factor equal to zero and finding the roots or solutions. The factor by grouping method allows for more flexibility in factoring quadratic expressions, especially when the coefficients are not easily identifiable or when there is a mix of positive and negative terms. This makes the process of solving quadratic equations more efficient and effective.

Related terms

Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers.

Greatest Common Factor (GCF): The greatest common factor (GCF) is the largest positive integer that divides each of the terms in a polynomial without a remainder.

Factoring: Factoring is the process of expressing a polynomial as a product of smaller polynomials or factors.