🔟elementary algebra review
7.1 Greatest Common Factor and Factor by Grouping
Last Updated on June 24, 2024
Factoring is a key skill in algebra, helping simplify complex expressions. The greatest common factor (GCF) is the largest factor shared by all terms in an expression. Finding the GCF allows us to break down polynomials into simpler forms.
Factoring by grouping is a technique for handling polynomials with four or more terms. This method involves pairing terms, factoring out common factors, and identifying shared binomials. These skills are crucial for solving equations and simplifying algebraic expressions.
Greatest Common Factor and Factoring
Greatest common factor in expressions
Top images from around the web for Greatest common factor in expressions
8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original
Is this image relevant?
Greatest Common Factor and Factor by Grouping – Intermediate Algebra View original
Is this image relevant?
8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original
Is this image relevant?
8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original
Is this image relevant?
Greatest Common Factor and Factor by Grouping – Intermediate Algebra View original
Is this image relevant?
1 of 3
Top images from around the web for Greatest common factor in expressions
8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original
Is this image relevant?
Greatest Common Factor and Factor by Grouping – Intermediate Algebra View original
Is this image relevant?
8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original
Is this image relevant?
8.4 Greatest Common Factor and Factor by Grouping – Introductory Algebra View original
Is this image relevant?
Greatest Common Factor and Factor by Grouping – Intermediate Algebra View original
Is this image relevant?
1 of 3
- Largest factor that divides all terms in an algebraic expression evenly without a remainder
- To find the GCF of an algebraic expression:
- Factor each term completely (using prime factorization)
- Identify common factors among all terms
- Choose factor with highest degree (variables) or largest value (numbers) as GCF
- Examples:
- : GCF is (common factor with highest degree)
- : GCF is (largest common factor)
Factoring polynomials with GCF
- Divide each term by GCF and factor it out
- Steps to factor polynomial using GCF:
- Identify GCF of polynomial
- Divide each term by GCF
- Write factored expression as product of GCF and quotient (result of division)
- Example:
-
- GCF is (common factor with highest degree)
- Dividing each term by GCF: and
- Factored expression: (factored form)
-
Factor by grouping method
- Used to factor polynomials with four or more terms
- Steps to factor by grouping:
- Group terms in pairs with a common factor
- Factor out GCF from each pair
- Identify common binomial factor in resulting expression
- Factor out common binomial
- Example:
-
- Group terms:
- Factor out GCF from each group:
- Identify common binomial factor:
- Factor out common binomial:
-
Understanding algebraic expressions
- Like terms: Terms with the same variables raised to the same powers
- Coefficient: The numerical factor of a term containing a variable
Key Terms to Review (17)
Prime Factorization: Prime factorization is the process of expressing a whole number as a product of its prime factors. It involves breaking down a number into a unique combination of prime numbers that, when multiplied together, result in the original number. This concept is fundamental to understanding various topics in elementary algebra, such as finding the greatest common factor, factoring polynomials, and simplifying rational expressions and square roots.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. It is a fundamental concept in elementary algebra that is applicable in various contexts, including whole numbers, fractions, and factoring.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the number or quantity that is applied to the variable, indicating how many times the variable is to be used in the expression.
Like Terms: Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be combined by adding or subtracting their coefficients, which are the numerical factors in front of the variables.
Distributive Property: The distributive property is a fundamental algebraic principle that allows for the simplification of expressions involving multiplication. It states that the product of a number and a sum is equal to the sum of the individual products of the number with each addend.
Polynomial: A polynomial is an algebraic expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers. Polynomials are fundamental building blocks in algebra and are central to many topics in elementary algebra.
Trinomial: A trinomial is a polynomial expression that contains three terms. It is a type of polynomial where the variable is raised to different powers, and the terms are connected by addition or subtraction operations.
Divisor: A divisor is a number or expression that divides another number or expression without leaving a remainder. It is a fundamental concept in mathematics, particularly in the context of division operations and finding the greatest common factor of polynomials.
Factor by Grouping: 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.
Factoring Out: Factoring out is a technique in algebra where a common factor is identified and extracted from a set of terms or expressions. This process simplifies the expression and can reveal the underlying structure of the problem, making it easier to solve.
Factoring by Grouping: 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.
Factoring Out the GCF: Factoring out the Greatest Common Factor (GCF) is a technique used in algebra to simplify polynomial expressions by identifying and extracting the largest factor that is common to all the terms in the expression. This process helps to break down complex expressions into more manageable components, making them easier to work with and understand.
Multiple: A multiple is a number that can be expressed as a product of a given number and a whole number. In other words, a multiple is the result of multiplying a number by a positive integer.
Factored Form: Factored form is a way of expressing a polynomial expression by breaking it down into a product of simpler factors. This representation can provide insights into the structure and properties of the polynomial, such as its roots and behavior.
Common factor: A common factor is a number or algebraic expression that divides two or more numbers or expressions evenly, leaving no remainder. Identifying common factors is essential for simplifying expressions, factoring polynomials, and performing operations with rational expressions. By recognizing these factors, one can break down complex problems into more manageable components, making mathematical operations clearer and easier to execute.
Factor: A factor is a number or expression that divides another number or expression evenly, meaning there is no remainder. In mathematics, understanding factors is crucial for simplifying expressions, solving equations, and performing polynomial operations. Factors can include integers, variables, or combinations of both, and they play a key role in finding the greatest common factor and in various strategies for breaking down polynomials into simpler components.
Binomial: A binomial is a polynomial that consists of exactly two terms separated by a plus or minus sign. These two terms can be made up of constants, variables, or both, and they can have different powers. Binomials are fundamental in algebra because they are often involved in operations such as addition, subtraction, multiplication, and factoring, and play a crucial role in simplifying expressions and solving equations.