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Section 6.1

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.

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Section 6.1

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

  2. 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

  3. 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 a factored form of x5 a factored form of factors factors factors Section 6.1

  4. Finding the Greatest Common Factor of a List of Integers • Greatest Common Factor • 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. Section 6.1

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. Factoring Out the Greatest Common Factor Factoring the GCF from a polynomial results in a product resembling the distributive property. GCF is 2 • 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. Section 6.1

  12. 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

  13. 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

  14. 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

  15. Factoring by Grouping • Factor Section 6.1

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