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, , 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 x 6 7x + 21y – 7 x 9 y 6 + x 3 y 5 – x 4 y 3 + x 3 y 3 9y 6 – 27y y 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 x 3 – 28x x – 21 6x – 42 + xy – 7y 4x 2 – 8xy – 3x + 6y