# MTH 091 Section 11.1 The Greatest Common Factor; Factor By Grouping.

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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 x 2 – 15x + 50 = (x – 5)(x – 10) Trinomial x 3 – 2x 2 + 5x – 10 = (x 2 + 5)(x – 2)Grouping 4x 2 – 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 x 3, x 2, x 5 p 7 q, p 8 q 2, p 9 q 3 32x 5, 18x 2 15y 2, 5y 7, -20y 3 40x 7 y 2 z, 64x 9 y

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 5x 2 + 10x 6 7x + 21y – 7 x 9 y 6 + x 3 y 5 – x 4 y 3 + x 3 y 3 9y 6 – 27y 4 + 18y 2 + 6 x(y 2 + 1) – 3(y 2 + 1) q(b 3 – 5) + (b 3 – 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 x 3 + 4x 2 + 3x + 12 16x 3 – 28x 2 + 12x – 21 6x – 42 + xy – 7y 4x 2 – 8xy – 3x + 6y

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