Transcript Ch6-Sec 6.2

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6.2 – Slide 1
Chapter 6
Factoring and Applications
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6.2 – Slide 2
6.2
Factoring Trinomials
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6.2 – Slide 3
6.2 Factoring Trinomials
Objectives
1.
2.
Factors trinomials with a coefficient of 1 for the
squared term.
Factor trinomials after factoring out the greatest
common factor.
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6.2 – Slide 4
6.2 Factoring Trinomials
Factoring Trinomials with a Coefficient of 1 for Squared Term
Example 1
Factor x2 + 8x + 12.
Look for two integers whose product is 12 and whose sum is 8.
Only positive integers are needed since all signs in x2 + 8x + 12
are positive.
From the list, 6 and 2 are the required
Factors Sums of
integers. Thus,
of 12
Factors
x2 + 8x + 12 = (x + 6)(x + 2)
12, 1
13
Check:
4, 3
7
2 + 2x + 6x + 12
(x
+
6)(x
+
2)
=
x
6, 2
8
= x2 + 8x + 12
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6.2 – Slide 5
6.2 Factoring Trinomials
Factoring Trinomials with a Coefficient of 1 for Squared Term
Note
In Example 1, the answer (x + 2)(x + 6) also could have
been written
(x + 6)(x + 2).
Because of the commutative property of multiplication,
the order of the factors does not matter. Always check
by multiplying.
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6.2 – Slide 6
6.2 Factoring Trinomials
Factoring Trinomials with a Coefficient of 1 for Squared Term
Example 2
Factor x2 – 9x + 18.
Find two integers whose product is 18 and whose sum is – 9.
Since the numbers we are looking for have a positive product
and a negative sum, we consider only pairs of negative integers.
Factors Sums of
of 18
Factors
–18, –1
–19
–9, –2
–6, –3
–11
–9
The required integers are –6 and –3, so
x2 – 9x + 18 = (x – 6)(x – 3)
Check:
(x – 6)(x – 3) = x2 – 3x – 6x + 18
= x2 – 9x + 18
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6.2 – Slide 7
6.2 Factoring Trinomials
Factoring Trinomials with a Coefficient of 1 for Squared Term
Example 3
Factor x2 – 3x – 10.
This time, we need to consider positive and negative integers
whose product is –10 and whose sum is –3.
Factors Sums of
of –10 Factors
10, –1
9
–10, 1
–9
5, –2
3
–5, 2
–3
Here –5 and 2 are the required integers.
x2 – 3x – 10 = (x – 5)(x + 2)
Check:
(x – 5)(x + 2) = x2 + 2x – 5x – 10
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= x2 – 3x – 10
6.2 – Slide 8
6.2 Factoring Trinomials
Factoring Trinomials with a Coefficient of 1 for Squared Term
Example 4
Factor x2 + 3x + 9.
Both factors must be positive to give a positive product and a
positive sum. We need to consider only positive integers.
Factors Sums of
of 9
Factors
9, 1
11
3, 3
6
None of the pairs of integers has a
sum of 3. Therefore, the trinomial
cannot be factored using only
integers.
This is a prime polynomial.
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6.2 – Slide 9
6.2 Factoring Trinomials
Factoring Trinomials with a Coefficient of 1 for Squared Term
Factoring x2 + bx + c
Find two integers whose product is c and whose sum is b.
1. Both integers must be positive if b and c are positive.
2. Both integers must be negative if c is positive and b is
negative.
3. One integer must be positive and one must be negative if
c is negative.
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6.2 – Slide 10
6.2 Factoring Trinomials
Factoring Trinomials with a Coefficient of 1 for Squared Term
Example 5
Factor x2 + 6xy – 7y2.
Though this trinomial has two variables, the process for
factoring it still works the same.
Here 7y and –y will work. So,
Factors Sums of
of –7y2 Factors
–7y, 1y
–6y
7y, –1y
6y
x2 + 6xy – 7y2 = (x + 7y)(x – y)
Check:
(x + 7y)(x – y) = x2 – xy + 7xy – 7y2
= x2 + 6xy – 7y2
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6.2 – Slide 11
6.2 Factoring Trinomials
Factoring Trinomials after Factoring out the GCF
Example 6
Factor 3a7 – 30a6 – 33a5.
First, factor out the greatest common factor, 3a5. Then, factor
the trinomial as usual.
3a7 – 30a6 – 33a5 = 3a5(a2 – 10a – 11)
= 3a5(a – 11)(a + 1)
Check:
3a5(a – 11)(a + 1) = 3a5(a2 – 10a – 11)
= 3a7 – 30a6 – 33a5
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6.2 – Slide 12