Factoring by Grouping

Definition

Factoring by grouping is a technique used to factor polynomials by first grouping the terms in the polynomial, then finding the greatest common factor (GCF) of each group, and finally combining the GCFs to obtain the final factorization. This method is particularly useful for factoring polynomials where the terms do not have a common factor.

5 Must Know Facts For Your Next Test

1. Factoring by grouping is a useful technique for factoring polynomials that do not have a common factor among all the terms.
2. The process involves dividing the polynomial into groups, finding the GCF of each group, and then combining the GCFs to obtain the final factorization.
3. Factoring by grouping is particularly effective for polynomials of the form $ax^2 + bx + cy + dz$, where $a$, $b$, $c$, and $d$ are coefficients.
4. This method can also be used to factor trinomials of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are coefficients.
5. Factoring by grouping is a crucial step in solving certain types of polynomial equations and simplifying algebraic expressions.

Review Questions

• Explain the step-by-step process of factoring a polynomial by grouping.
  • To factor a polynomial by grouping, follow these steps: 1. Divide the polynomial into groups, usually based on the variables or the coefficients. 2. Find the greatest common factor (GCF) of each group. 3. Factor out the GCF from each group. 4. Combine the factored groups using the distributive property to obtain the final factorization.
• How does factoring by grouping differ from other factoring techniques, such as factoring trinomials of the form $ax^2 + bx + c$?
  • Factoring by grouping is distinct from factoring trinomials of the form $ax^2 + bx + c$ in that it is used for polynomials that do not have a common factor among all the terms. While factoring trinomials involves finding two numbers whose product is $ac$ and whose sum is $b$, factoring by grouping focuses on identifying the GCF of each group of terms and then combining these GCFs to obtain the final factorization. This makes factoring by grouping particularly useful when the polynomial does not have an obvious common factor.
• Explain how factoring by grouping can be applied to solve polynomial equations and simplify algebraic expressions.
  • Factoring by grouping is a crucial step in solving certain types of polynomial equations and simplifying algebraic expressions. By factoring the polynomial, you can rewrite it as a product of simpler factors, which can then be used to solve the equation or simplify the expression. For example, if you have a polynomial equation like $4x^2 + 12x + 6 = 0$, you can factor it by grouping to obtain $(2x + 3)(2x + 2) = 0$, which can then be solved by setting each factor equal to zero. Similarly, factoring by grouping can be used to simplify complex algebraic expressions by breaking them down into smaller, more manageable components.

Related terms

Greatest Common Factor (GCF): The largest positive integer that divides each of the given integers without a remainder.

Polynomial: An expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers.

Factorization: The process of expressing a polynomial or number as a product of its factors.