Section 6.3&4
Factoring Trinomials, Part 2

Review from last homework assignment:
Factor the polynomial 3x 6 + 30x 5 + 72x 4 .
First we factor out the GCF . ( Always check for this first!)
3 x 6 + 30 x 5 + 72 x 4 = 3 x 4 ( x 2 + 10 x + 24)
Then we factor the trinomial.
Positive factors of 24 Sum of Factors
1, 24 25
2, 12
3, 8
4, 6
14
11
10
So 3 x 6 + 30 x 5 + 72 x 4 = 3 x 4 ( x 2 + 10 x + 24)
= 3 x 4 ( x + 4)( x + 6).

For this problem, there is no common factor that we can pull out of all three terms of the trinomial.
Question: Does this automatically mean that this polynomial is prime (can’t be factored) just because we can’t take out the 2 in front of the x 2 term??
Answer: Not necessarily. In fact, this polynomial DOES factor.
The factoring is (2x + 1)(x – 13). How would you check to see if this factoring is correct?

• In all of yesterday’s homework problems except #32, the leading coefficient of the trinomial was 1 , or became 1 after the GCF was factored out, as in the review problem we just did.
•
But what if the leading coefficient is NOT 1 , and is still larger than
1 after any GCF is factored out, like in problem #32?
•
In some cases, the resulting trinomial is prime, but in many other cases, it can be factored into two binomials.
Example: Factor 3x 2 + 14x + 8 .
There is no GCF that can be factored out.
However, this polynomial is NOT prime.
The factors of this trinomial are (3x + 2)(x+4). (check)
But how do we figure this out to start with?

Back to the problem: Factor 3x 2 + 14x + 8 .
• We’re going to show you a method for these kinds of problems that may be easier for you than the “guess and check” method shown in some of the textbook/online examples.
• This method is sometimes called the “ British Method ”, and it starts with splitting the middle term of the trinomial into two parts, so that we can use factoring by grouping on the resulting four-term polynomial.
First: multiply the first and last coefficients of the trinomial together:
3*8 = 24
Now look for two factors of 24 that add up to 14 (the middle number):
24 = 3*8, but 3 + 8 = 11
24 = 6*4, but 6 + 4 = 10
24 = 2*12, and 2 + 12 = 14 , which is what we’re looking for.
Now split the middle term, 14x, into two pieces, 2x + 12x
This now gives 3x 2 + 2x + 12x + 8
Now we have a FOUR-TERM polynomial. What does this suggest??

You got it: Factoring by grouping!
3x 2 + 2x + 12x + 8
Factoring the first pair gives: 3x 2 + 2x = x(3x + 2)
Factoring the 2 nd pair gives: 12x + 8 = 4(3x + 2)
Putting it all back together now gives: x(3x+2) + 4(3x + 2) = (3x + 2)(x+4) Done!
Don’t forget to always check your answer by multiplying it back out. (You should get in the habit of doing this on your homework so you don’t forget to do it on quizzes.)

Example Factor the polynomial 25 x 2 + 20 x
First, always check for a GCF .
There isn’t one, so now we will multiply the first and last coefficients together:
25*4 = 100
Now find two factors of 100 that add up to 20:
2* 50? Nope (2 + 50 = 52)
5*20? Nope (5 + 20 = 25)
4*25? Nope (4 + 25 = 29)
10*10? Bingo!
(10 + 10 = 20)
Use this to split the middle term ( 20x ) into two pieces:
25x 2 + 10x + 10x + 4.
Now factor by grouping. Answer: (5x + 2)(5x + 2) (check it!)
What’s a better way to write this? (5x + 2) 2
+ 4.

In the examples we’ve done so far, it looks like the “factoring by grouping” method always works for a trinomial with a leading coefficient other than one that can’t be factored away.
But consider this example: Factor 3x 2 + 15x + 8 .
First: multiply the first and last coefficients of the trinomial together:
3*8 = 24
Now look for two factors of 24 that add up to 15. product: sum:
24 = 1*24 1+ 24 = 25
24 = 2*12 2 + 12 = 14
24 = 3*8
24 = 6*4
3 + 8 = 11
6 + 4 = 10
There are no other ways to factor 24, and none of our factor pairs add up to 15, so this polynomial is prime .

Example
Factor the polynomial 21 x 2 – 41 x + 10.
1. Multiply 21*10 = 210
2. Find 2 factors of 210 that add up to -41
[Note: 210 = 2*3*5*7]
-10*-21? What do these factors add up to?
-2*-105?
-5*-42?
-15*-14?
-3*-70?
-6*-35? BINGO!
Now factor by grouping: 21x 2 - 6x – 35x+10
3x(7x – 2) -5(7x – 2) = (7x +2)(3x – 5) NOW CHECK IT!

Example
Factor the polynomial 3 x 2 + 20 x – 4.
Step 1: 3*-4 = -12
Step 2: possible factors of -12:
3*-4 sum = -1
-3*4
-1*12
1*-12
-2*6 sum = 1 sum = 11 sum = -11 sum = 4
2* -6 sum = -4
There are no other possible ways to factor -12, and none of the combinations add up to 20, so this is a PRIME polynomial (it can’t be factored).

Example
Factor the polynomial 6 x 2 y 2 – 10 xy 2 – 4 y 2 .
Remember to ALWAYS check for a GCF first. The
GCF of the three terms of this polynomial is 2y 2 , so we factor that out first:
6 x 2 y 2 – 10 xy 2 – 4 y 2 = 2 y 2 (3 x 2 – 5 x – 2)
Now factor (3x 2 -5x -2) by the “factoring by grouping” method shown in the previous examples.
Final answer: 2y 2 (3x + 1)(x - 2) (remember to check!)

The assignment on t oday’s material ( HW 6.3/4 ) is due at the start of the next class session.
on the Chapter 6 material is next week , so if you start to get behind of the homework, try to find time to come into the open lab for help NOW instead of waiting till a day or two before the test.
Homework Questions?
Use the Open Lab!
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OPEN
We expect all students to stay in the classroom to work on your homework till the end of the 55minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice quiz/test.