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Section 5.5 (The Greatest Common Factor and Factoring by Grouping) The greatest common factor (GCF) among 2 or more numbers is the largest number that is a factor of all the numbers The GCF of monomials is the product of o The GCF of the coefficients o The variable(s) common in the monomials (use smallest exponent present for that variable) Examples: Find the GCF of 15x3, 30x2, and 40x5 Find the GCF of 16p3y, -10py, 40p2y2 and 12p Factoring a polynomial is essentially the reverse of multiplying monomials / polynomials To factor a given polynomial, find the GCF of all terms, and write the answer as a product of the GCF and the remaining terms (5x2 + 15 = (5)x2 + (5)3 = 5(x2 + 3)) Examples: Factor (check results) 7x3 + 21x2 2 3 1 2 4 x - x + 3 3 3 15x4 – 9x3 – 21x 16p6q4 + 32p5q3 – 48pq2 Certain polynomials can also be factored by a method of grouping Examples: Factor (by grouping) x7(x + 5) + 8 (x + 5) = (x + 5)(x7 + 8) = (x + 5)(x7 + 8) 3x5(2x - 7) – (2x – 7) = A good place to start with a 4-term polynomial is by grouping the first 2 terms together, factoring, and then factoring the last 2 terms x3 + 6x2 + 4x + 24 = (x3 + 6x2) + 4x + 24 = x2(x + 6) + 4x + 24 = x2(x + 6) + 4(x + 6) = (x2 + 4)(x + 6) 20x3 – 12x2 + 35x – 21 a2b2 + a2 - 3b2 - 3 20p3 – 4p2 -25p + 5