Transcript 7.5 Factoring x^2+bx+c and 7.6 Factoring 〖ax〗^2+bx+c

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7.5 Factoring 𝑥
+ 𝑏𝑥 + 𝑐
and
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7.6 Factoring 𝑎𝑥 + 𝑏𝑥 + 𝑐
STUDENTS WILL BE ABLE TO FACTOR TRINOMIALS IN THE FORM 𝑎𝑥 2
+ 𝑏𝑥 + 𝑐 .
Steps to follow:
Step 1) Always look for a gcf first.
Step 2) Fill in X organizer.
Step 3) Split the middle term.
Step 4) Factor by grouping.
Step 5) Rewrite answer in factored form.
ac
b
Remember you can always check your work by multiplying the two factors back
together.
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Example 1: Factor 𝑥 + 10𝑥 + 16
1) There is no gcf.
2)
16
2
8
10
3) Split middle term:
𝑥 2 + 8𝑥 + 2𝑥 + 16
4) Factor by grouping:
(𝑥 2 +8𝑥) + 2𝑥 + 16
x(x + 8) + 2(x + 8)
5) Rewrite answer in factored form:
(x + 2)(x + 8)
Check by multiplying:
𝑥 2 + 8𝑥 + 2𝑥 + 16
𝑥 2 + 10𝑥 + 16
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Example 1: Factor 𝑥 − 8𝑥 + 12
1) There is no gcf.
2)
12
-2
-6
-8
3) Split middle term:
𝑥 2 − 6𝑥 − 2𝑥 + 12
4) Factor by grouping:
(𝑥 2 −6𝑥) + −2𝑥 + 12
x(x - 6) - 2(x - 6)
5) Rewrite answer in factored form:
(x - 2)(x - 6)
Check by multiplying:
𝑥 2 − 6𝑥 − 2𝑥 + 12
𝑥 2 − 8𝑥 + 12
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You try: Factor 𝑥 + 7𝑥 + 6
1) There is no gcf.
2)
6
6
1
7
3) Split middle term:
𝑥 2 + 𝑥 + 6𝑥 + 6
4) Factor by grouping:
(𝑥 2 +𝑥) + 6𝑥 + 6
x(x + 1) + 6(x + 1)
5) Rewrite answer in factored form:
(x + 6)(x + 1)
Check by multiplying:
𝑥 2 + 𝑥 + 6𝑥 + 6
𝑥 2 + 7𝑥 + 6
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Example 3: Factor 𝑥 + 4𝑥 − 21
1) There is no gcf.
2)
-21
-3
7
4
HINT: when c is negative you
must come up with a positive
and negative number!
3) Split middle term:
𝑥 2 − 3𝑥 + 7𝑥 − 21
4) Factor by grouping:
(𝑥 2 −3𝑥) + 7𝑥 − 21
x(x - 3) + 7(x - 3)
5) Rewrite answer in factored form:
(x + 7)(x - 3)
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You try: Factor 𝑥 + 13𝑥 − 30
1) There is no gcf.
2)
-30
-2
15
13
3) Split middle term:
𝑥 2 + 15𝑥 − 2𝑥 − 30
4) Factor by grouping:
(𝑥 2 +15𝑥) + −2𝑥 − 30
x(x + 15) -2(x + 15)
5) Rewrite answer in factored form:
(x - 2)(x + 15)
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Example 4: Factor 5𝑥 + 15𝑥 + 10
1) 5(𝑥 2 + 3𝑥 + 2)
2)
5(𝑥 2 + 2𝑥 + 𝑥 + 2)
2
2
1
3
3) Split middle term:
4) Factor by grouping:
5[(𝑥 2 +2𝑥) + 𝑥 + 2 ]
5[x(x + 2) + 1(x + 2)]
5) Rewrite answer in factored form:
5(x + 1)(x + 2)
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Example 5: Factor 4𝑥 + 13𝑥 + 3
1) There is no gcf.
2)
4𝑥 2 + 12𝑥 + 𝑥 + 3
12
12
1
13
3) Split middle term:
4) Factor by grouping:
(4𝑥 2 +12𝑥) + 𝑥 + 3
4x(x + 3) + 1(x + 3)
5) Rewrite answer in factored form:
(4x + 1)(x + 3)
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Example 6: Factor −4𝑥 − 8𝑥 + 5
1) –(4𝑥 2 + 8𝑥 − 5)
2)
-20
10
-2
8
3) Split middle term:
−(4𝑥 2 + 10𝑥 − 2𝑥 − 5)
4) Factor by grouping:
−(4𝑥 2 + 10𝑥) + −2𝑥 − 5
-[2x(2x + 5) - 1(2x + 5)]
5) Rewrite answer in factored form:
When “a” is negative factor out -1 first!
-(2x - 1)(2x + 5)
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You try: Factor 8𝑥 − 56𝑥 + 48
1) 8(𝑥 2 − 7𝑥 + 6)
2)
6
-1
-6
-7
3) Split middle term:
8(𝑥 2 − 6𝑥 − 𝑥 + 6)
4) Factor by grouping:
8(𝑥 2 − 6𝑥) + −𝑥 + 6
8[x(x – 6) - 1(x - 6)
5) Rewrite answer in factored form:
8(x - 1)(x - 6)
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You try: Factor 2𝑥 − 7𝑥 + 5
1) There is not gcf.
2)
10
-2
-5
-7
3) Split middle term:
2𝑥 2 − 5𝑥 − 2𝑥 + 5
4) Factor by grouping:
(2𝑥 2 −5𝑥) + −2𝑥 + 5
x(2x – 5) - 1(2x - 5)
5) Rewrite answer in factored form:
(x - 1)(2x - 5)