Extra Examples (a=1)

Review
Factoring Trinomials
in the form ax2 + bx + c, (a = 1)
FACTORING REVERSES MULTIPLICATION.
For example: What can you multiply to get x2?
Answer: x times x
Therefore, x2 in factored form is x(x).
List pairs of factors of 12.
1 & 12, 2 & 6, 3 & 4
Objective: To factor trinomials.
Steps for Factoring
x2 + bx + c
Watch and think.
Factor x2 – 9x + 14 .
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2.
3.
4.
5.
6.
Factor out GCF.
Find factors of “c” that combine to give you “b”.
When “c” is positive, the sign in each binomial is
the same as “b”.
If “c” is negative, the sign in each binomial is
different.
Write as the product of 2 binomials. GCF( )( )
Use FOIL or the Distributive Prop. to check.
Examples with positive “c” term
a) g2 + 7g + 10
b) v2 + 21v + 20
c) a2 + 13a + 30
d) q2 – 15q + 36
Check for GCF. (There is no GCF.)
For the first term, x2 = (x
)(x
)
Because “c” is positive, add factors of “c” to get “b”, which is 9.
Factors of 14 are 1 & 14, 2 & 7. Which pair will combine to = 9?
(x 2) (x 7)
Now the signs: The last term is positive which means the signs
are the same. Look at the middle term. The signs need to be
negative.
Final answer: (x - 2) (x - 7)
Check with FOIL: x2 – 7x – 2x + 14 which equals x2 – 9x + 14,
the original problem.
Examples – Solutions (positive “c”
means signs will be same)
a) g2 + 7g + 10 Factors of 10: 1x10, 2x5
2 + 5 = 7 so factored form = (g + 2)(g + 5)
b) v2 + 21v + 20 Factors of 20: 1x20, 2x10, 4x5
1 + 20 = 21 so factored form = (v + 1)(v + 20)
c) a2 + 13a + 30 Factors of 30: 1x30, 2x15, 3x10, 5x6
3 + 10 = 13 so factored form = (a + 3)(a + 10)
d) q2 – 15q + 36 Factors of 36: 1x36, 2x18, 3x12, 4x9
3 + 10 = 13 so factored form = (q + 12)(q + 3)
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Examples
Examples -- Solutions
e) b2 + 11b + 10
e) b2 + 11b + 10 = (b + 10)(b + 1)
f) k2 – 10k + 25
f) k2 – 10k + 25 = (k – 5)(k – 5)
g) x2 – 11xy + 18y2
g) x2 – 11xy + 18y2 = (x – 2y)(x - 9y)
h) n2 – 9mn + 20n2
h) n2 – 9mn + 20n2 = (n - 4m)(n – 5m)
Watch and think.
Factor 3x3 – 3x2 – 18x .
What if “c” is negative?
Every example we’ve done so far has had
a positive “c” term. Therefore, we’ve
combined the factors of “c” to get the “b”
term and both terms had same sign.
Now we’ll look at when “c” is negative.
What do you think will change in how we
pick the factors to use?...Instead of terms
having the same sign, the terms will have
different signs.
Examples with negative “c” term
a) m2 + 8m – 20
b) p2 – 3p – 40
c) y2 – y – 56
d) n2 – 5n - 24
Check for GCF. (GCF = 3x.)
Factor it out: 3x(x2 – x – 6).
Factor the first term in the trinomial, 3x(x
)(x
)
Because “c” is negative, one factor will be positive and one
will be negative. (Think of subtracting the factors since signs
are different.)
Factors of 6 are 1 & 6 and 2 & 3. Which pair will subtract to
give 1?
3x(x 2) (x 3)
Now the signs: The last term is negative which means the
signs are different. Look at the middle term. Which number
needs to be negative to give -1?
Final answer: 3x(x + 2) (x - 3)
Check with FOIL: 3x(x2 – 3x + 2x – 6) which gives
2
3x(x –x- 6), which equals 3x3 – 3x2 – 18x, the original
problem.
Examples -- Solutions
a) m2 + 8m – 20 = (m + 10)(m – 2)
Factors of 20: 1x20, 2x10, 4x5; 10 and -2 combine
to = 8 (which is “b”)
b) p2 – 3p – 40 = (p – 8)( p + 5)
Factors of 40: 1x40, 2x20, 4x10, 5x8; -8 and 5
combine to = -3
c) y2 – y – 56 = (y – 8)(y + 7)
Factors of 56: 1x56, 2x28, 4x14, 7x8; -8 and 7
combine to = -1
d) n2 – 5n – 24 = (n – 8)(n + 3)
2
Examples
x2
+ 11xy +
Examples
24y2
e)
f) v2 + 2vw – 48w2
g) m2 – 17mn – 60n2
h) d2 + 17dg – 60g2
Note: If there is a variable squared at the end
of the trinomial, simply add that variable to
the end of each term in factored form.
e) x2 + 11xy + 24y2 = (x + 3y)(x + 8y)
Factors of 24: 1x24, 2x12, 3x8, 4x6; 3 and 8
combine to = 11 (which is “b”)
f) v2 + 2vw – 48w2 = (v - 6w)(v + 8w)
Factors of 48: 1x48, 2x24, 3x16, 4x12, 6x8; -6 and
8 combine to = 2
g) m2 – 17mn – 60n2 = (m - 20n)(m + 3n)
Factors of 60: 1x60, 2x30, 3x20, 4x15, 6x10; -20
and 3 combine to = -17
h) d2 + 17dg – 60g2 = (d + 20g)(d - 3g)
SUMMARY
Factoring x2 + bx + c
Find factors of “c” that combine to give
you “b”.
2. If “c” is positive, the terms have the
same sign.
3. If “c” is negative, the terms will have
different signs.
4. Use FOIL to check.
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