Transcript MTH 098 - Shelton State Community College

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
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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
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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
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x3 + 4x2 + 3x + 12
16x3 – 28x2 + 12x – 21
6x – 42 + xy – 7y
4x2 – 8xy – 3x + 6y