The Greatest Common Factor and Factoring by Grouping Section 6.1 The Greatest Common Factor and Factoring by Grouping

The Greatest Common Factor and Factoring by Grouping Find the greatest common factor of a list of integers. Find the greatest common factor of a list of terms. Factor out the greatest common factor from a polynomial. Factor a polynomial by grouping. Section 6.1

The Greatest Common Factor and Factoring by Grouping Factored Form A number or expression is said to be factored when written as a product of factors. a factored form of 28 factors a factored form of x5 factors a factored form of factors Section 6.1

Finding the Greatest Common Factor of a List of Integers When given a set of two or more numbers, the largest natural number that evenly divides all the numbers in the set is called the greatest common factor. To find the GCF using factor pairs: list all factor pairs for each number select the largest number that appears in both lists Find the GCF of 45 and 75. Find the GCF of 36 and 42. 45 1 3 15 5 9 75 1 3 25 5 15 Section 6.1

Finding the Greatest Common Factor of a List of Integers Find the GCF for the expressions The GCF of a list of common variables raised to powers is the smallest exponent in the list. We can extend this idea by using what is known as the prime factorization. 4 factors of x in common, or Section 6.1

Finding the Greatest Common Factor of a List of Integers To find the GCF using prime factorization: Find the prime factorization of each number using a factor tree. Determine which factors the numbers have in common. The GCF will be the product of each common prime factor. Find the GCF for the numbers 72 and 90 72 90 8 9 10 9 2 4 3 3 3 3 2 5 2 2 Section 6.1

Finding the Greatest Common Factor of a List of Integers To find the GCF using prime factorization: Find the prime factorization of each number using a factor tree. Determine which factors the numbers have in common. The GCF will be the product of each common prime factor. Find the GCF for the numbers 72 and 90 32 and 33 14, 24, and 60 54 and 99 Section 6.1

Finding the Greatest Common Factor of a List of Terms In general, the GCF of a list of monomials, is the product of the GCF for the coefficients and the variables. Find the GCF of the monomials Section 6.1

Factoring Out the Greatest Common Factor The GCF of a polynomial is the GCF of the individual monomial terms. Find the GCF of Section 6.1

Factoring Out the Greatest Common Factor Factored Form A number or expression is said to be factored when written as a product of factors. Factoring is answering, “what can I multiply to get the given expression?” Your answer will look like a multiplication problem like the ones from Chapter 5! Section 6.1

Factoring Out the Greatest Common Factor Factoring the GCF from a polynomial results in a product resembling the distributive property. To factor a monomial GCF out of a given polynomial Find the GCF of the terms in the polynomial. Write the terms as a product containing the GCF. Factor out the GCF (un-distribute). The given polynomial is written as a product of the GCF and the result of dividing the polynomial by the GCF. GCF is 2 Section 6.1

Factoring Out the Greatest Common Factor Factor the GCF A GCF of 6ac is fine, but we really don’t like to see (-a… If the first term is negative, it is best to take out a negative GCF, even if it is just -1. Section 6.1

Factoring by Grouping Factoring the GCF is only one stage of factoring. Sometimes a polynomial can be factored further. Polynomials with four terms are factored with a process called grouping. Section 6.1

Factoring by Grouping To factor by grouping Factor the GCF from all terms if possible Group the terms into pairs Factor the GCF from each pair Factor out the common binomial factor from each group. If the remaining binomials are not common: Try rearranging the terms before grouping. You did not remove the correct GCF. The polynomial cannot be factored. Section 6.1

Factoring by Grouping Factor Section 6.1