Transcript Chapter 6, Section 1 - City Colleges of Chicago

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6.1 – Slide 1
Chapter 6
Factoring and Applications
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6.1 – Slide 2
6.1
Factors; The Greatest
Common Factor
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6.1 – Slide 3
6.1 Factors; The Greatest Common Factor
Objectives
1.
2.
3.
4.
Find the greatest common factor of a list of
numbers.
Find the greatest common factor of a list of variable
terms.
Factor out the greatest common factor.
Factor by grouping.
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6.1 – Slide 4
6.1 Factors; The Greatest Common Factor
Finding the Greatest Common Factor of a List of Numbers
The greatest common factor (GCF) of a list of integers
is the largest common factor of those integers. This
means 6 is the greatest common factor of 18 and 24,
since it is the largest of their common factors.
Note
Factors of a number are also divisors of the number.
The greatest common factor is the same as the
greatest common divisor.
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6.1 – Slide 5
6.1 Factors; The Greatest Common Factor
Finding the Greatest Common Factor of a List of Numbers
Example 1
Find the greatest common factor for each list of numbers.
(a) 36, 60
First write each number in prime factored form.
36 = 2 · 2 · 3 · 3
60 = 2 · 2 · 3 · 5
Use each prime the least number of times it appears in all the
factored forms. Here, the factored forms share two 2’s and one 3.
Thus,
GCF = 2 · 2 · 3 = 12.
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6.1 – Slide 6
6.1 Factors; The Greatest Common Factor
Finding the Greatest Common Factor of a List of Numbers
Example 1 (continued)
Find the greatest common factor for each list of numbers.
(b) 18, 90, 126
Find the prime factored form of each number.
18 = 2 · 3 · 3
90 = 2 · 3 · 3 · 5
126 = 2 · 3 · 3 · 7
All factored forms share one 2 and two 3’s. Thus,
GCF = 2 · 3 · 3 = 18.
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6.1 – Slide 7
6.1 Factors; The Greatest Common Factor
Finding the Greatest Common Factor of a List of Numbers
Example 1 (concluded)
Find the greatest common factor for each list of numbers.
(c) 48, 61, 72
48 = 2 · 2 · 2 · 2 · 3
61 = 1 · 61
72 = 2 · 2 · 2 · 3 · 3
There are no primes common to all three numbers, so the
GCF is 1.
GCF = 1
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6.1 – Slide 8
6.1 Factors; The Greatest Common Factor
Finding the Greatest Common Factor for Variable Terms
Note
The exponent on a variable in the GCF is the least
exponent that appears on that variable in all the terms.
Example 2
Find the greatest common factor for each list of terms.
(a) 12x2, –30x5
12x2 = 2 · 2 · 3 · x2
–30x5 = –1 · 2 · 3 · 5 · x5
First, 6 is the GCF of 12 and –30. The least exponent on x is 2
(x5 = x2 · x3). Thus,
GCF = 6x2.
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6.1 – Slide 9
6.1 Factors; The Greatest Common Factor
Finding the Greatest Common Factor for Variable Terms
Example 2 (concluded)
Find the greatest common factor for each list of terms.
(b) –x5y2, –x4y3, –x8y6, –x7
–x5y2,
–x4y3,
–x8y6,
–x7
There is no y in the last term. So, y will not appear in the GCF.
There is an x in each term, and 4 is the least exponent on x. Thus,
GCF = x4.
Note In a list of negative terms, sometimes a negative
common factor is preferable (even though it is not the
greatest common factor). In (b) above, we might prefer
–x4 as the common factor.
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6.1 – Slide 10
6.1 Factors; The Greatest Common Factor
Finding the Greatest Common Factor for Variable Terms
Finding the Greatest Common Factor (GCF)
Step 1 Factor. Write each number in prime factored form.
Step 2 List common factors. List each prime number or
each variable that is a factor of every term in the
list. (If a prime does not appear in one of the prime
factored forms, it cannot appear in the greatest
common factor.)
Step 3 Choose least exponents. Use as exponents on
the common prime factors the least exponents from
the prime factored forms.
Step 4 Multiply. Multiply the primes from Step 3. If there
are no primes left after Step 3, the greatest
common factor is 1.
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6.1 – Slide 11
6.1 Factors; The Greatest Common Factor
Factor Out the Greatest Common Factor
CAUTION
The polynomial 3m + 12 is not in factored form when
written as the sum
3 · m + 3 · 4.
Not in factored form
The terms are factored, but the polynomial is not.
The factored form of 3m + 12 is the product
3(m + 4).
In factored form
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6.1 – Slide 12
6.1 Factors; The Greatest Common Factor
Factor Out the Greatest Common Factor
Example 3
Factor out the greatest common factor.
(a) 24x5 – 40x3 = 8x3(3x2) – 8x3(5)
GCF = 8x3
= 8x3(3x2 – 5)
Note If the terms inside the parentheses still have a common
factor, then you did not factor out the greatest common factor
in the previous step.
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6.1 – Slide 13
6.1 Factors; The Greatest Common Factor
Factor Out the Greatest Common Factor
Example 3 (concluded)
Factor out the greatest common factor.
(b) 4x6y4– 20x4y3 + x2y2 = x2y2(4x4y2) – x2y2(20x2y) + x2y2(1)
= x2y2(4x4y2 – 20x2y +1)
CAUTION
Be sure to include the 1 in a problem like Example 3(b).
Check that the factored form can be multiplied out to
give the original polynomial.
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6.1 – Slide 14
6.1 Factors; The Greatest Common Factor
Factor Out the Greatest Common Factor
Example 4
Factor – 3x5 – 15x3 + 6x2.
– 3x5 – 15x3 + 6x2 = – 3x2(x3 + 5x – 2)
GCF = – 3x2
Note
Whenever we factor a polynomial in which the coefficient of
the first term is negative, we will factor out the negative
common factor, even if it is just – 1.
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6.1 – Slide 15
6.1 Factors; The Greatest Common Factor
Factor Out the Greatest Common Factor
Example 5
Factor out the greatest common factor.
w2(z4– 3) + 5(z4 – 3)
Here, the binomial z4 – 3 is the GCF.
w2(z4– 3) + 5(z4 – 3) = (z4– 3)(w2 + 5)
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6.1 – Slide 16
6.1 Factors; The Greatest Common Factor
Factor By Grouping
Example 6
Factor by grouping.
6x + 4xy – 10y – 15
If we leave the terms grouped as they are, we could try factoring
out the GCF from each pair of terms.
6x + 4xy – 10y – 15 = 2x(3 + 2y) – 5(2y + 3)
This works, showing a common binomial of 2y + 3 in each term.
6x + 4xy – 10y – 15 = 2x(2y + 3) – 5(2y + 3)
= (2y + 3)(2x – 5)
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6.1 – Slide 17
6.1 Factors; The Greatest Common Factor
Factor Out the Greatest Common Factor
CAUTION
Be careful with signs when grouping in a problem
like Example 6. It is wise to check the factoring in the
second step before continuing.
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6.1 – Slide 18
6.1 Factors; The Greatest Common Factor
Factor By Grouping
Factoring a Polynomial with Four Terms by Grouping
Step 1 Group terms. Collect the terms into two groups so
that each group has a common factor.
Step 2 Factor within groups. Factor out the greatest
common factor from each group.
Step 3 Factor the entire polynomial. Factor a common
binomial factor from the results of Step 2.
Step 4 If necessary, rearrange terms. If Step 2 does not
result in a common binomial factor, try a different
grouping.
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6.1 – Slide 19
6.1 Factors; The Greatest Common Factor
Factor By Grouping
Example 7
Factor by grouping.
10a2 – 12b + 15a – 8ab
Working as before, we get
10a2 – 12b + 15a – 8ab = 2(5a2 – 6b) + a(15 – 8b)
This does not work. These two factored terms have no binomial
in common. So, we will group another way.
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6.1 – Slide 20
6.1 Factors; The Greatest Common Factor
Factor By Grouping
Example 7 (concluded)
Factor by grouping.
10a2 – 12b + 15a – 8ab = 10a2 – 8ab + 15a – 12b
= 2a(5a – 4b) + 3(5a – 4b)
This works, showing a common binomial of 5a – 4b in each
term. Thus,
10a2 – 12b + 15a – 8ab = (5a – 4b)(2a + 3)
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6.1 – Slide 21