Download 10-5 Factoring Quadratic Trinomials

10-5 Factoring Quadratic
Trinomials
Homework
Any Questions?
Factoring quadratic trinomials is FUN!! It is kind of like
solving a mathematical riddle or puzzle.
When you factor a trinomial there are 2 clues to take into account.
1. Operator Signs
Trinomial
Binomial
+ +
( + )( + )
- +
( -)( - )
- -
( - )( + )
+ -
( + )( - )
•The first sign remains the SAME the second sign is the result of
multiplying the signs together (a positive # x a positive # = a
positive #; a negative # x a positive # = a negative #...)
2. Factors of c that add/subtract to equal b
Example 1:
Factor the trinomial
x2 + 3x + 2
Step 1: Check the signs
Step 2: Factors of “c” that add up to “b”
Step 3: Factor your trinomial to 2 binomials
Step 4: Check with FOIL
Let’s try the steps…
Step 1: + +
(+)(+)
Step 2: Factors of 2 are 2 & 1 and 2 + 1 = 3
Step 3: (x + 2)(x + 1)
Step 4: x2 + 2x + x + 2 , combine like terms to get…
x2 + 3x + 2
Cool Fact of the Day
Some species of fish have
voices.
5
Example 2: Factor the trinomial x2 – 2x – 8
Step 1: (-)(+)
Step 2: Factors of -8 which add up to 2 are -4 & 2
Step 3: (x – 4) (x + 2)
Step 4: x2 + 2x – 4x – 8
x2 – 2x – 8 original trinomial
Student of the Day
Factoring ax2 + bx + c
when a = 1
What two numbers
multiply to give you
8, but add to give
you 6?
To FACTOR a trinomial means to write
it as the product of two binomials.
Factor x2 + 6x + 8
What
twotwo
numbers
Write
sets of
multiply to give you
parenthesis and
the last number…
filltoin
the
and add
give
you
the middle
number?
numbers.
(x + 2)) (x + 4))
Ex: 1
Factor x2 - 3x + 2
(x - 2)) (x - 1))
What two numbers
multiply to give you
the last number…
and add to give you
the middle number?
Ex: 2
Factor x2 - 2x - 8
(x - 4) (x + 2)
Ex: 3
Factor x2 - 5x - 14
(x - 7)) (x + 2))
Ex: 4
Factor x2 - 16x + 64
(x - 8)) (x - 8))
Same thing
2
as (x - 8)
Ex: 5
Factor x2 - x - 42
(x + 6))
(x - 7)) (x
Solve the equation by
factoring
Ex: 6
x2 - 8x + 12 = 0
(x
)(x
)=0
x-6=0
or
x–2=0
x=6
or
x =2
Solve the equation by
factoring
2
Ex: 7
x - 7 = - 6x
x2 + 6x - 7 = 0
(x
)(x
)=0
x-1=0
or
x+7=0
x=1
or
x = -7
Homework
Complete Sum and Product Puzzles
plus Pg. 539 #5-19 odd
Now, factoring can become a little more challenging when “a” is not 1.
You will use the strategy of Guess and Check to find the factors.
Your goal is to find a combination of factors of a and c
so that the inner and outer products add to the middle
term of bx.
Factors of “a”
ax2 + bx + c = ( _x + _ )( _x + _ )
Factors of “c”
Example 3: Factor the Trinomial: 6x2 – 7x – 5
You will have to use Guess and Check for this trinomial because “a” is
not one anymore.
Guess & Check
(x + 1)(6x – 5)
6x2 + x – 5, does not work
(x + 5)(6x – 1)
6x2 + 29x – 5
(2x + 1)(3x – 5)
6x2 – 7x – 5, yes we have found the answer!
Example 4: Factor the Trinomial: 6x2 – 7x – 5
New method
1.
(6x
)(6x
2. (6x – )(6x
3. (6x– )(6x +
) First term, without the exponent, goes in each
binomial
) First sign goes in the first binomial
) Both signs multiplied together, goes in the
second binomial (negative times a negative
equals a positive)
4. Multiply a and c together = 30 then find all the factor
combinations for 30… 1,30 2,15 3,10 5,6
5. If the signs in the two binomials are the same use the two
factors that add to b
If the signs in the two binomials are different use the two
factors that subtract to b
6.
(6x–10)(6x +3) Put the larger factor in the first binomial and the
other in the second binomial.
7. Factor each binomial (6x–10)/2 (6x +3)/3
(3x-5) (2x+1)
Student of the Day
Many trinomials cannot be factored with integer
coefficients and there is a way to check for factorability.
Using the discriminant, b2 – 4ac, to check for factorability.
If your answer is a perfect square, it can be factored, but
if it is not a perfect square then you must use other
methods.
Example 4:
2x2 + 3x – 6
b2 – 4ac = 32 – 4 (2)(-6) = 57
Therefore, this trinomial cannot be factored.
Cool Photo of the Day
New Invention
23
Cool Photo of the Day
New Invention
24
Cool Photo of the Day
New Invention
25
Cool Photo of the Day
New Invention
26
Factor the trinomial.
Use the discriminant to decide whether the
polynomial can be factored with integer
coefficients. If it can, factor it.
Homework
Homework: Pg. 539 29-33 odd Pg.
556 #37-46