MTH 091

Section 11.1The 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 Factorx2 – 15x + 50 = (x – 5)(x – 10) Trinomialx3 – 2x2 + 5x – 10 = (x2 + 5)(x – 2) Grouping4x2 – 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 coefficients2.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 GCFTerm1 + 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