Download Section 1.6 Factoring Trinomials

Section 1.6
Factoring Trinomials
Pages 28 - 31
What are trinomials?
•
A trinomial will have 3 terms
•
There are four steps used to factor each
of them
1.
2.
3.
4.
Check for a common monomial factor
Look for special cases
Factor by trial and error
Continue factoring as long as possible
1. Check for a common factor
• For each trinomial, it simplifies the factoring
process if you look for a common factor
• What is the common factor in this expression?
2 x2 y – 8 x y + 8 y
• Since each term is divisible by 2, the 2 is
common
• Each term also contains a y, so the common
monomial is 2y
• When factored: 2y ( x2 – 4x + 4 )
What is the common factor?
• find the common factor, then check your
answer
3x2y3 – 6x5y + 12x3z
answer:
3x2
14a2b2c2 + 7a3b3c3d + 21ab2c
answer: 7ab2c
2. Check for special cases
• Once all of the common factors have been
removed, look for special cases
• The only special case we’ll look at here is
the perfect square trinomial
a2 + 2ab + b2
2. Check for special cases
• If there is a perfect square, the first and
last terms must be perfect squares
x2 + 4x + 4
• The first term ( x2 ) and last term ( 4 ) are
both perfect squares
2. Check for special cases
• a perfect square will also need to have a
specific middle term
• take the square root of the first term,
multiply it by the square root of the last
term, then multiply that by 2
2. Check for special cases
•
•
•
•
x2 + 4x + 4
in this equation, the square root of the first
term is x
the square root of the last term is 2
when multiplied, x ( 2 ) ( 2 ) = 4x
since this matches the middle term in the
original expression, this is a perfect square
2. Check for special cases
x2 + 4x + 4
• the factors for this will follow the pattern:
(x+2)2
• the first number inside of the brackets is the
square root of the first term, the second number
is the square root of the last term
• the sign inside of the brackets will match the
sign of the second term
Which of these are perfect
squares?
4x2 + 16x + 25
8x2 + 16x + 9
x2 - 12x + 36
81x2 + 38x + 100
9x2 + 25x + 4
16x2 + 32x + 3
2x2 + 10x + 25
4x2 + 12x - 9
3. factor by trial and error
• if the expression is still not factored
completely, you must factor by trial and
error
3. factor by trial and error
•
•
•
•
3x2 - 10x + 8
to factor, you need all of the factors of the
first and last terms
first term: 1 x 3
last 1 x 8 , 2 x 4
these need to be combined until the
correct set of factors is obtained
3. factor by trial and error
3x2 - 10x + 8
• remember, if the last term is positive, both
of the factors have the same sign
– if the second term is negative, then both are
negative
– if it is positive, then both are positive
• if it is negative, then they have opposite
signs
3. factor by trial and error
3x2 - 10x + 8
• since there are only 2 factors for the first
term, we can easily guess the first part of
the 2 factors
( 3x - ) ( x )
• we also know that the signs inside of the
brackets are negative, since the sign of
the second term was negative, but the
third term was positive
3. factor by trial and error
3x2 - 10x + 8
( 3x - ) ( x )
• Now we must choose two factors of 8 and
see if we obtain the correct answer
– This might take a few attempts.
3. factor by trial and error
3x2 - 10x + 8
( 3x - 1 ) ( x - 8 )
When multiplied, we get:
3x2 - 24x - x + 8
This is not the correct answer.
3. factor by trial and error
3x2 - 10x + 8
( 3x - ) ( x )
• We don’t need to do all of the steps of
FOIL, we only need the middle term, so
we only multiply the outside and inside
terms
3. factor by trial and error
3x2 - 10x + 8
( 3x - 4 ) ( x - 2 )
• Lets try two different factors.
• When multiplied, we get:
-6x - 4x
Combined together, we have -10x
These are now the correct factors
4. Continue factoring until reach
factor is prime
• Remember to apply as many steps as
needed until the trinomial is factored as
much as possible