Greatest Common Factor and Factoring by Grouping 6.1
Find the Greatest Common Factor (GCF) of a Set of Terms Factored form: A number or an expression written as a product of factors. Examples: 2 x + 8 = 2( x + 4) x 2 + 5 x + 6 = ( x + 2)( x + 3) Greatest common factor (GCF) of a set of terms: A monomial with the greatest coefficient and degree that evenly divides all of the given terms.
Method To find the GCF of two or more monomials, 1. Write the prime factorization in exponential form for each monomial. 2. Write the GCF’s factorization by including the prime factors (and variables) common to all the factorizations, each raised to its smallest exponent in the factorizations. 3. Multiply the factors in the factorization created in step 2. Note: If there are no common prime factors, then the GCF is 1.
Exercise 1 Find the GCF [06] 6m 4 n 9 , 15mn 5 [12] 8(m – n), 11(m – n), m – n
A Method to Find the Prime Factorization of a Constant Determine how many times two can go into the constant, then the number of threes, fives, sevens, elevens, etc. Example: Factor 420 2 ) 420 2 ) 210 3 ) 105 5 ) 35 7 ) 7 1 There are 2 twos, 1 three, 1 five, 1 seven 420 = 2 2 3 1 5 1 7 1 = 2 2 3 5 7
Factor a Monomial GCF Out of the Terms of a Polynomial The method to factor a monomial GCF out of the terms of a polynomial is: 1. Find the GCF of all the terms in the polynomial. 2. Rewrite the polynomial as a product of the GCF and the quotient of the polynomial and the GCF.
Exercise 2 Factor: [20] -2x 2 y 2 - 8x 3 y + 2xy [30] a( b – 5 ) + 7 ( b – 5 )
Factor Polynomials by Grouping Method to factor a four-term polynomial by grouping: 2x 2 - 4x - 4x + 8 1. Factor out any monomial GCF (other than 1) that is common to all four terms. 2. Group together pairs of terms and factor the GCF out of each pair. 3. If there is a common binomial factor, then factor it out. 4. If there is no common binomial factor, then interchange the middle two terms and repeat the process. If there is still no common binomial factor, then the polynomial cannot be factored by grouping.
Exercise 3 Factor: [50] 15 c 3 – 5 c 2 d + 6 cd 2 – 2 d 3 [62] 12 ac + 12 cx - 3 ac 2 - 3 c 2 x [**] 2x 3 – y 3 – 2x 2 y + xy 2
Greatest Common Factor and Factoring by Grouping 6.1
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