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MTH 091 Section 11.1 The Greatest Common Factor; Factor By Grouping What Does It Mean To Factor? • To factor a number means to write it as the product of two or more numbers: 24 = 6 x 4, or 15 = 5 x 3, or 30 = 2 x 3 x 5 • To factor a polynomial means the same thing— that is, to write it as a product: 6x – 15 = 3(2x – 5) Greatest Common Factor x2 – 15x + 50 = (x – 5)(x – 10) Trinomial x3 – 2x2 + 5x – 10 = (x2 + 5)(x – 2) Grouping 4x2 – 25 = (2x + 5)(2x – 5) Difference of Squares Finding the GCF of a List of Numbers 1. Find the prime factorization for each number (use a factor tree). 2. Circle the common factors in each list of numbers. 3. Multiply the circled numbers together. This is your GCF. Find the GCF • 36, 90 • 30, 75, 135 • 15, 25, 27 Find the GCF of a List of Terms 1. Find the GCF of the coefficients (see previous slide). 2. For common variables: choose the smallest exponents. Find the GCF • • • • • x3, x2, x5 p7q, p8q2, p9q3 32x5, 18x2 15y2, 5y7, -20y3 40x7y2z, 64x9y Now What? • Once you find the GCF, you factor it out of each term in your polynomial: Polynomial = GCF(Leftovers) 1. Divide the coefficients 2. Subtract the exponents • If you multiply your GCF by your leftovers, you should get your original polynomial back. Factor Out the GCF • • • • • • • 42x – 7 5x2 + 10x6 7x + 21y – 7 x9y6 + x3y5 – x4y3 + x3y3 9y6 – 27y4 + 18y2 + 6 x(y2 + 1) – 3(y2 + 1) q(b3 – 5) + (b3 – 5) Factor By Grouping • Used to factor a polynomial with four terms. 1. Look at the first two terms and factor out their GCF. 2. Now look at the last two terms and factor out their GCF Term1 + Term2 + Term3 + Term4 = GCF1(Leftovers) + GCF2(Leftovers) = (Leftovers)(GCF1 + GCF2) 3. Rearranging the four terms is allowed. Factor By Grouping • • • • x3 + 4x2 + 3x + 12 16x3 – 28x2 + 12x – 21 6x – 42 + xy – 7y 4x2 – 8xy – 3x + 6y