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Surprise! Surprise!
Factoring pattern number
FIVE
Quadratic Trinomials
• With integral coefficients
What does that mean?
Sometimes a trinomial will have a number
before the quadratic term
Factoring pattern for
ax² + bx + c
Some examples of such trinomials are:
5y² - 17 y + 6
2b² + 13b – 24
4y² + 4y - 3
2a² - 15 a – 27
Yikes!!!!!
• Those trinomials
looks really scary!!
• There must be a
pattern
3x² -x - 4
• The constant term is negative. So we know
we must have one negative and one
positive number in our binomials.
• The first step is to list the possible factors
of the quadratic term and the possible
factors of the constant term
3x² - x - 4
• Possible factors of 3
x²
3x x
• Possible factors of – 4
4 -1
-1
4
-2 2
Now What???
• Now you begin
testing all the possible
arrangements of those
factors
• Remember one
number is positive
and one negative
•Are we having fun yet?
Look for the linear term
Remember it is -x
•
•
•
•
•
•
(3x + 4)(x – 1)
(3x- 4) (x + 1)
(3x – 1) (x + 4)
(3x + 1) (x – 4)
(3x + 2) (x – 2)
(3x – 2) ( x + 2)
• What arrangement
works?
Wow! This takes work
•
•
•
•
•
•
(3x + 4) (x - 1)
(3x - 4)(x + 1)
(3x – 1) (x + 4)
(3x + 1) (x – 4)
(3x + 2) (x – 2)
(3x – 2) (x +2)
•
•
•
•
•
•
-3x + 4x = x
3x - 4x = - x
12x - x = 11x
-12x + x = -11x
-6x + 2x = -4x
6x – 2x = 4x
SOOOOOOO
3x² - x - 4
Can be factored into these binomials
(3x - 4) ( x + 1)
Use Foil to check
That was sure lots of fun!
Remember these steps
• Notice the sign of the constant and the linear
term. This will tell you if you need two positive
numbers or two negative numbers or one of each
• List the possible factors of the quadratic term
• List the possible factors of the constant
• TEST the possibilities. A chart may be helpful
And remember
• Practice makes perfect!!!!!
Are you ready to try one?
2a² -15 a +27
Let’s see….first I….
Notice that both terms
will have to be
negative
Now I list all those factors
2a² - 15 a + 27
• Factors of 2a²
2a
a
• Now the factors of 27
(Keeping in mind that
since the linear term is
negative both factors
must be negative)
-27
-1
-1
-27
-9
-3
-3
-9
OK this is great
• List the possible
combinations
(2a- 27) (a –1)
(2a –1)( a – 27)
(2a – 9) (a – 3)
(2a – 3) (a – 9)
• Now what would each
linear term be (We
hope one is –15a!)
-2a – 27a = - 29a
-54a – a = -55a
-6a – 9a = - 15a
-18a –3a = -21 a
VICTORY!
2a² - 15 a + 27
• What were those
winning factors???
• Oh yeah!
• (2a- 9) (a –3)
• Don’t forget FOIL
• Yep it works
• 2a ² - 6a – 9a + 27
2a² - 15a + 27
Assignment
• Open your math book to page 220 and do
1 to 200 inclusive
WHAT? You’ re whining?
OKAY OKAY you can use a