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M95. Mod 2. Sec. 5.6 A: Factoring Trinomials:
page 1
I. Review FOIL: multiplying to get a trinomial, changing from a product to a sum
Ex. 1: Multiply:
 x  4 x  5 =
a) What are the factors of ex. 1?
b) What are the terms of the answer?
Ex. 2: Multiply:
 x 1 x  4 =
a) What are the factors of ex 2?
b) What are the terms of the answer?
II. Factoring by Trial-and-Error or Guess-and-Check: Factoring is undistributing.
It is writing the sum as a product.
A. When the coefficient on the squared term is 1:
Factor the following:
Ex. 3: x 2  3 x  2
Notice: The sign on the last term
of the trinomial is positive—the
signs in the binomials must be the same.
Ex. 4: x 2  8 x  15
Notice: The sign on the last term
of the trinomial is positive—the
signs in the binomials must be the same.
M95, Sec. 5.6a, pg.2
Ex. 5: y 2  2y 15
Notice: The sign on the last term
of the trinomial is negative—the
signs in the binomials must be different.
Ex. 6: x 2  2xy  8y 2
Notice: The sign on the last term
of the trinomial is negative—the
signs in the binomials must be different.
Ex. 7: a2  6a  9
Ex. 8: x  8 x  15
2
Ex. 9: x  10 x  24
2
Ex. 10: a  8a  16
2
M95, Sec. 5.6a, pg.3
B. When the coefficient on the squared term is NOT 1: Look for a GCF first.
Ex. 11: 5y 2  5y  30
Ex. 12: 2z2  12z  16
Ex. 13: 3m4  18m3  15m2
Ex. 14: 6 x  24 x  30 x
3
2
Ex. 15: 3a  6a  6
2
Ex. 16:
M95, Sec. 5.6a, pg.4
III. When the coefficient on the squared term is not 1, and there is no common
factor, try guess and check or factoring by grouping.
Ex. 16: 2y 2  5y  3
Ex. 17: 3 p2  13p  30
Ex. 18: 6 x 2  x  12
Ex. 19: 4z 2  17z  4
Ex. 20: 3 x  13 x  4
2
Ex. 21: 5 x  13 x  6
2
M95. Mod 2. Sec. 5.6B: More Factoring of Trinomials:
page 1
I. When the coefficient on the squared term is not 1, and there is no common factor,
try guess and check. Use the sign clues from the previous lesson to make good
educated guesses.
Ex. 1:
Ex. 2:
Ex. 3:
Ex. 4:
Ex. 5: 12x  11x  2
2
Ex. 6: 14 x  x  3
2
M95, Sec. 5.6B pg 2
Ex. 7: 6 x  13 x  8
2
II. When do we know to quit trying to factor a polynomial? In the case of quadratic
trinomials, when the discriminant is a perfect square, it can be factored with integers.
When a polynomial cannot be factored, we call it prime.
d =
III. The "AC" or "First-Last" method:
Ex. 8: 8 x  40 xy  50y
2
Ex. 9:
Ex.10:
2
M95, Sec. 5.6B pg 3
IV. Reviewing Patterns:
Ex. 11: Multiply:
Ex. 12: Factor:
Ex. 13: Multiply:
Ex. 14: Factor:
Hint: rewrite 27 as
M95, Sec. 5.6B pg 4
Ex. 15: Multiply:
Ex. 16: Factor:
Ex. 17: Factor:
Ex. 18: Factor:
Ex. 19: Factor:
(Check with multiplication)
M95. Mod 2. Sec. 5.7: Factoring by Special Products
I. Review: Factor the following
Ex. 1: 4z 2  17z  4
Ex. 2: 4 x 2  32x  60
Ex. 3: 24z3  8z 2  80z
Ex. 4: 3  5 x  2x 2
II. The Difference of Two Squares (The Difference of Squares):
A. Multiply:  a  b a  b  (the sum and difference of the same terms)
B. a2  b2
1.
2.
3.
4.
Binomial
Each term is a perfect square
It is the difference (subtraction)
It will factor into:  a  b a  b 
5. Note: the sum of two squares is prime! i.e. a  b is prime.
2
C. Factor the following:
Ex. 5: y 2  144
2
Ex. 6: 16 x 2  100
Ex. 7: 9 x 2  4y 2
M95, Sec. 5.7, pg 2
Ex. 8: 64z 4  49z 2
Ex. 9: t 4  9
Ex. 10: x 4  81t 4
Ex. 11: 4 x 3  8 x 2  x  2
III. Perfect Square Trinomials: (comes from squaring a binomial)
A. Multiply:
 3 x  2y 
2
B. Perfect Square Trinomials:
1. It is a trinomial
2. The first and last terms are perfect squares (and both positive!)
3. The middle term is twice the product of the original terms in the
binomial
4. It will be of the form: A2  2AB  B2 or A2  2AB  B 2
5. It will factor into:
C. Factor the following:
Ex. 12: 4z 2  4z  1
 A  B
2
or
A B
2
Ex. 13: 9y 2  6y  4
M95, Sec. 5.7, pg 3
Ex. 14: x 2  2x  1
Ex. 15: x 2  20 x  100
Ex. 16: 9z3  6z 2  z
Ex. 17: 50 x 2  60 x  18
Ex. 18: 64 x 2  16 x  1
Ex. 19: x 2  2x  1
IV. A Factoring Strategy:
1. Look for a common factor (gcf). If there is one, factor it out first.
2. Is the polynomial a binomial (two terms)? If so, is it the difference of two
squares? a2  b2   a  b a  b  . Remember that the sum of two squares is
prime.
3. Is the polynomial a trinomial (three terms)?
2
a. Is it a perfect square? A2  2 AB  B 2   A  B 
b. If it is not a perfect square, use trial-and-error.
4. Does the polynomial contain four or more terms? Try grouping.
5. Is the polynomial completely factored? Check each factor.
V. More Practice : Factor completely.
Ex. 20: 4 x 2  22x  10
Ex. 21: 75a2  48
Ex. 22: 10 x 3  90x 2  200x
Ex. 23: n3  2n 2  n  2
Ex. 24: 2r 3  12r 2t  18rt 2
Ex. 25: 4r 4  t 6
Ex. 26: 24  x 2  5 x
Ex. 27: 4a2  27a  18
Ex. 28: 25 x 2  x  36
Ex. 29: 2x 2  11x  20
Ex. 30: x 2  25
Ex. 31:
Ex. 32:
M95, Sec. 5.7, pg 4