Please open your laptops and pull up Quiz 6.2.You will have 10 minutes for this quiz.

If you have any time left after finishing the quiz problems,

CHECK YOUR FACTORING ANSWERS before you submit the quiz.

Please

CLOSE

YOUR LAPTOPS,and turn off and put away your

cell phones,

and get out your note-taking materials.

Section 6.3&4

Factoring Trinomials, Part 2

Review from yesterday’s homework:

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, 8 114, 6 10

So 3x6 + 30x5 + 72x4 = 3x4(x2 + 10x + 24) = 3x4(x + 4)(x + 6).

But what about a problem like #32 from the 6.2 HW that was due today?

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 x2 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 3x2 + 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 3x2 + 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 = 24Now 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 3x2 + 2x + 12x + 8

Now we have a FOUR-TERM polynomial. What does this suggest??

You got it: Factoring by grouping!

3x2 + 2x + 12x + 8 Factoring the first pair gives: 3x2 + 2x = x(3x + 2) Factoring the 2nd 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

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:

25x2 + 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

Factor the polynomial 25x2 + 20x + 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 3x2 + 15x + 8.

First: multiply the first and last coefficients of the trinomial together:

3*8 = 24Now look for two factors of 24 that add up to 15. product: sum:

24 = 1*24 1+ 24 = 2524 = 2*12 2 + 12 = 1424 = 3*8 3 + 8 = 11

24 = 6*4 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 21x2 – 41x + 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: 21x2 - 6x – 35x+10

3x(7x – 2) -5(7x – 2) = (7x +2)(3x – 5) NOW CHECK IT!

Example

Factor the polynomial 3x2 + 20x – 4.Step 1: 3*-4 = -12

Step 2: possible factors of -12:

3*-4 sum = -1

-3*4 sum = 1

-1*12 sum = 11

1*-12 sum = -11

-2*6 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 6x2y2 – 10xy2 – 4y2.

Remember to ALWAYS check for a GCF first. The GCF of the three terms of this polynomial is 2y2, so we factor that out first:

6x2y2 – 10xy2 – 4y2 = 2y2(3x2 – 5x – 2)

Now factor (3x2 -5x -2) by the “factoring by grouping” method shown in the previous examples.

Final answer: 2y2(3x + 1)(x - 2) (remember to check!)

REMINDERS:The assignment on today’s material (HW 6.3/4)

is due at the start of the next class session.

Lab hours in 203:

Mondays through Thursdays

8:00 a.m. to 6:30 p.m.Please remember to sign in on the Math 110 clipboard

by the front door of the lab

Test 3 on the Chapter 6 material is a week from tomorrow (Thursday, April 17), 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.

You may now OPEN your LAPTOPS

and begin working on the homework assignment.

We expect all students to stay in the classroom to work on your homework till the end of the 55-minute 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.