Section 6.2
Factoring Trinomials, Part 1
Review problem from Section 6.1:
(Factoring out the GCF and
factoring 4-term polynomials by grouping)
Factor completely:
20 x y  30 x y  4 x y  6 xy
3
2
Answer:
2
2 xy (2 x  3)(5 x  1)
How would you check this?
(Because you WOULD do that, wouldn’t you???)
Section 6.2
Factoring Trinomials, Part 1
Recall by using the FOIL method that
(x + 2)(x + 4)
= x2 + 4x + 2x + 8
= x2 + 6x + 8
So to factor x2 + 6x + 8 into (x + __ ) (x + __ ), note that
6 is the sum of the two numbers 4 and 2,
and 8 is the product of the two numbers.
So we’ll be looking for 2 numbers whose product is 8 and
whose sum is 6.
Note: there are fewer possible pairs of numbers for the
product than for the sum, so that’s why we start there first.
Example
Factor the polynomial x2 + 13x + 30.
Since our two numbers must have a product of 30 and a
sum of 13, the two numbers must both be positive.
Positive factors of 30
Sum of Factors
1, 30
1+30=31
2, 15
1+15=17
3, 10
3+10=13
Note, there are other factors (like 6*5), but once we
find a pair that works, we do not have to continue
searching.
Now check answer by multiplying
two factors to see if you get
So x2 + 13x + 30 = (x + 3)(x + 10). the
back to the original trinomial.
Example
Factor the polynomial x2 – 11x + 24.
Since our two numbers must have a product of 24 and a
sum of -11, the two numbers must both be negative.
Negative factors of 24
Sum of Factors
-1, -24
-25
-2, -12
-14
-3, -8
-11
So x2 – 11x + 24 = (x – 3)(x – 8).
Now check it!
Example
Factor the polynomial x2 – 2x – 35.
Since our two numbers must have a product of -35 and a
sum of -2, the two numbers will have to have
different signs.
Factors of -35
Sum of Factors
-1, 35
34
1, -35
-34
-5, 7
2
5, -7
-2
So x2 – 2x – 35 = (x + 5)(x – 7).
Check it!
Example
Factor the polynomial x2 – 6x + 10.
Since our two numbers must have a product of 10 and a
sum of -6, the two numbers will have to both be
negative.
Negative factors of 10
Sum of Factors
-1, -10
-11
-2, -5
-7
Now we have a problem, because we have exhausted
all possible choices for the factors, but have not found
a pair whose sum is -6.
So x2 – 6x +10 is not factorable
and we call it a prime polynomial.
Example Factor the polynomial x2 – 11xy + 30y2.
We look for two terms whose product is 30x2y2 and
whose sum is –11xy. The two terms will have to both
be negative.
Note: each term will contain the variable y, for the sum
to be –11xy.
Negative factors of 30x2y2 Sum of Factors
-xy, -30xy
-31xy
-2xy, -15xy
-17xy
-3xy, -10xy
-13xy
-5xy, -6xy
-11xy
So x2 – 11xy + 30y2 = (x – 5y)(x – 6y).
Example Factor the polynomial 3x6 + 30x5 + 72x4
First we factor out the GCF. (Always check for this first!)
3x6 + 30x5 + 72x4 = 3x4(x2 + 10x + 24)
Then we factor the trinomial.
Positive factors of 24
Sum of Factors
1, 24
25
2, 12
14
3, 68
11
4,
10
So 3x6 + 30x5 + 72x4 = 3x4(x + 4)(x + 6).
REMINDER:
• On a test or quiz, if you have time left after finishing all the
problems, you should always check your factoring results
by multiplying the factored polynomial to verify that it is
equal to the original polynomial.
• Many times you can detect computational errors or errors
in the signs of your numbers (i.e those pesky “dumb
mistakes”…) by checking your results.
• Practice doing this on at least a few homework problems
before you hit “Check Answer”, just to make sure you
really do know how to check your answers when it comes
time for the quiz.
REMINDER:
The assignment on today’s material (HW 6.2) is due at
the start of the next class session.
Homework Questions?
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