Question

@mathmale How does one begin factoring x^2 - x - 6?"

Answer 1

We have a couple of different approaches to consider.
With this particular problem, P, I'd suggest you list the possible factors of -6. I would list them as

Answer 2

{1, 2, 3, 6, -1,-2, -3, -6}.

Answer 3

Which two of these combine to give you -1, the coefficient of the middle term of the expression you've typed in?

Answer 4

3 - 2 = 1 would be wrong; we wanted -1, not 1.

Answer 5

hm 1 and -2

Answer 6

Exactly right! -2 + 1 = -1.
It's a bit of a jump from here to there, but we can write the factors of your expression by stealing those 2 factors of -6 and writing the factors as (x-[1]) and (x-[-2]).

These simplify to (x-1) and (x+2).
That's it.

Check these factors by multiplying them together, using the FOIL method of multiplying binomials.

Answer 7

After a while you get the feel for this and the work becomes a lot easier and faster.

Answer 8

what does it equal ? i got a different answer

Answer 9

I've probably goofed somewhere; pls give me a moment to check your work and mine.

We're so brilliant that one of us is bound to be right, and that could be you. :)

Answer 10

:D

Answer 11

plohrr:

hm 1 and -2

I was way off in not recognizing that (1)(-2) do not multiply out to -6, even tho they do add up to -1.

So I'll need to ask y ou to try again. Find other possible pairs in

Answer 12

{1, 2, 3, 6, -1,-2, -3, -6}. that do combine to yield -1 and whose product is -6, the last term in your trinomial expression.

Answer 13

ohh ok.

Answer 14

3 and -2!!

Answer 15

Do those multiply out to -6?

Answer 16

do those combine to yield -1?

Answer 17

I'm making myself a sandwich for lunch, so there will be brief delays here.

Answer 18

aw man wait

Answer 19

OK, try again; see what you can come up with that satisfies both criteria.

Answer 20

2 and -3!

Answer 21

satisfied that those 2 criteria are satisfied?

Answer 22

If you are, then 2 => (x-2) (note the opposite signs) and -3=> (x-[-3]) => (x+3).

Multiply these two binomials together and see wehther you end up with your original trinomial expression.

Answer 23

yes it does

Answer 24

Isn't that great? This means you have successfully factored x^2 - x - 6!

Answer 25

YESSS

Answer 26

Do y ou have a list of homework problems? If so, would you mind choosing one more trinomial factor from that list and attempt factoring it?

Answer 27

give me another example !

Answer 28

(Big pat on back (yours))

Answer 29

i don't unfortunately :/

Answer 30

2x^2 + x -3

I need to double check this (for ease in factoring), but believe it's OK.

Have you any idea of how to start? Earlier I asked you to write possible factors of -6, and we got {1, 2, 3, 6, -1,-2, -3, -6}.

Now, the approach is a bit different: Multiply the 1st and last coefficients together (obtaining -6) and do exactly the same thing. Answer: {1, 2, 3, 6, -1,-2, -3, -6}.. I didn't plan this in this way; it just happened.

Answer 31

2*(-3) = -6, right? And -6 is the product of the first coefficient (2) and the last (-3). Agreed or not?

Answer 32

yes agreed

Answer 33

but now we need to get two factors that combine for -1 and multiply to -3 right

Answer 34

BRB

Answer 35

You might just luck out and find a possible factor right away. Or you might have to try dozens before you find the right ones (that is, factors that really do multiply into 2x^2 + x -3).
Try guessing: write a possible factor; we'll check whether or not it actyually is a factor.

Then we might move on to a more systematic way of doing this.

Answer 36

I need to know:
Have you had any experience with synthetic division? long division? factoring by grouping?

Answer 37

yes i do

Answer 38

im not comfortable with them tho

Answer 39

Have you formed one possible factor that we could check?

Answer 40

x + 2

Answer 41

Lucky you. That happens to be a factor! But we must demonstrate that it is indeed a factor of 2x^2 + x -3. How would you do that if left to your own devices?

Answer 42

umm. we would have to find another factor right ?

Answer 43

Yes, but it'd make3 a bit more sense to write, "we need to verify that x+2 is indeed a factor of 2x^2 + x -3.

Of these two, which is the least scary? synthetic division or long division?

Answer 44

synthetic division

Answer 45

i am not that familiar

Answer 46

I was hoping you'd say that. it's so much easier than long div.


-2 | 2 1 -3

------------
2

Answer 47

Please multiply: -2 times that 2 at the very bottom. Hope this looks familair.

Answer 48

it doesn't at all lol

Answer 49

but -4

Answer 50

Write the -4 under the 1 but above the ---- line.

Answer 51

-2 | 2 1 -3
-4
------------
2

Answer 52

-2 | 2 1 -3
-4
------------
2

Answer 53

Combine that 1 and -4.

-2 | 2 1 -3
-4
------------
2

Answer 54

Write the result (-3) below the ----- line and to the right of the 2 at the bottom.

Answer 55

-2 | 2 1 -3
-4
------------
2 -3

Answer 56

Wonderful.
-2 | 2 1 -3
-4
------------
2 -3

Muliply that -2 and that -3 together; write the product under the -3 next to the -4.

Answer 57

-2 | 2 1 -3
-4 6
------------
2 -3

Answer 58

-2 | 2 1 -3
-4 6
------------
2 -3

Answer 59

Add together that -3 and that 6 in the rightmost column.

Answer 60

-2 | 2 1 -3
-4 6
------------
2 -3 3

Answer 61

Great.
Here's the rule:
If, at the end of synth. div., the sum of the numbers in the last column is zero, then the divisor (here, -2) is a root of the original expression.
If NOT zero (which is the case here)

Answer 62

then the divisor (-2 here) is NOT a root of the original expression and the associated factor (x+2) is not actually a factor. I told you before that x+2 was a factor and that you had been lucky, but I was WRONG. Sorry ab out that. We'll

Answer 63

check x=1 instead of -2 this time:

1 | 2 1 -3

-----------------
2

Answer 64

Know what to do next?

Answer 65

Multiply that 1 (the divisor) and the 2 at the very bottom together. Write this product under the 2nd 1, just above the ----- line.

Answer 66

um. is there a way to memorize all the division stuff ? it seems random

Answer 67

It's not at all random. I'll help you become familiar with it.

1 | 2 1 -3
2
-----------------
2

Answer 68

Please add up the digits in the next-to last column. Copy the work above as you did before and write in the sum of 1 and 2 in the appropriate place.

Answer 69

1 | 2 1 -3
2
-----------------
2 3

Answer 70

Great; now mult the divisor (1) and this last sum (3) together and write their product under the -3.

Answer 71

1 | 2 1 -3
2 3
-----------------
2 3

Answer 72

Now, as before, add up the last column. What do y ou get?

Answer 73

1 | 2 1 -3
2 3
-----------------
2 3 0

Answer 74

Here's the rule:
If, at the end of synth. div., the sum of the numbers in the last column is zero, then the divisor (here, -2) is a root of the original expression.

Answer 75

so what is your conclusionj? is 1 a root or is it not? Is (x-1) a factor, or is it not?

Answer 76

no it is not they add to 5

Answer 77

Plohrr: Add the numbers in the last COLUMN, not the numbers in the last ROW. :)

Answer 78

OHH ok yes it is then

Answer 79

so, you see, if you add up the last col and you get a final sum of zero, then yes, 1 is a root of the polynomial expression and (x-1) is a factor of that expression. There you go.

I still think this is easier than long division, but long d. will work equally well.

Answer 80

ok, so now we just guess the other factor and do synthtic div. until we find it ?

Answer 81

I'm going to need to stop, as much as I'd love to continue. I've been on the computer more or less continuously for the past 7 hours. If there's any way in which you could take notes from or copy our conversation, I urge you to do so; it's so important to have something to study and review later.

Answer 82

ok ! thanks for your help :D i learned a lot

Answer 83

Actually, you're almost done!!

1 | 2 1 -3
2 3
-----------------
2 3 0

Drop the final 0 in the last row:

1 | 2 1 -3
2 3
-----------------
2 3

Think of the 2 and the 3 as coefficients within another factor of the original expression:
3 is the constant coeff.; 2 is the coeff of x: 2x+3=0 results in x = -3/2 (a root) and (2x+3) is a factor!

Answer 84

Please make up a list of things that you'd like to discuss next time. Of course there is more to factoring than this, but this was a good, strong introduction to the concepts involved.

Answer 85

:) bye !

Answer 86

all the best to you. Bye!