9-6a-24a^2
Do I have to rearrange this?

- anonymous

9-6a-24a^2
Do I have to rearrange this?

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- anonymous

It's a factoring question.

- hedgepig

oh. yeah
-(24a^2+6a-9)
take out the common factor 3
-3(8a^2+2a-3)
so this is a "hard factoring problem"

- hedgepig

try... 2 and 4 for a, and -3 and 1 for the constant.

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## More answers

- anonymous

I'm really confused by this. Would appreciate greatly any help. :-)

- hedgepig

-3(2a-1)(4a+3)

- anonymous

I'm a bit confused about what you did, hedgepig. Perhaps I need to review your text for a bit. If you could, please do stay. :-)

- anonymous

Okay, so what is the way to figure out the order?

- hedgepig

first, you want to factor out the largest constant.

- anonymous

What comes first? A, AB, or B? This is just in general.

- anonymous

Is a squared before a?

- hedgepig

9-6a-24a^2
factor out the largest number you can first.

- anonymous

Yes, but I guess I just need to know, in general, the rule for how you order things. Because, I have more problems, of course.

- anonymous

So, it's just, if you wouldn't mind, I'd like to know how to properly order this trinomials.

- anonymous

It's so important.

- hedgepig

there are multiple ways to process this, but, we can do that too. Order from highest degree to lowest.

- anonymous

But I don't what's highest. Is a^2 higher or lower than a?

- hedgepig

is it? what is the degree of a^2?

- anonymous

a^2 is a x a, but I don't know whether it would come before or after a.

- anonymous

If I were to guess, I would say after.

- anonymous

I would say a exponents, followed by B. I would guess ab would come before B, too.

- hedgepig

from what I gather you're missing knowledge from polynomials and terminology. Do you what the degree of a term is?

- anonymous

Unfortunately, I am not very knowledgeable on the terminology.

- hedgepig

Ok. You NEED to know this stuff. that being said, it shouldn't be too hard.
let's do some examples:
6x^3
23x^5
x^3
8x
9
-5x^2
identify the degree of each term.

- anonymous

I would guess the degree is the exponent.

- hedgepig

yes.

- anonymous

So, degrees are synonymous with exponents?

- hedgepig

no.

- hedgepig

what we are referring to is, as far as I can tell, a specific realm - the realm of polynomials.

- hedgepig

to answer your question, yes, degree is synonymous with exponents in the context of polynomials, but I am not sure I would say x^(-pi) would have degree -pi.

- anonymous

Okay.

- anonymous

In the poly and trinomial universe, degrees and exponents are the same thing.

- anonymous

That's the universe my head is in right now. :P

- hedgepig

that's not a problem in this context, but I would really advise you think broader than this if you want success mathematically.

- anonymous

Now, I really do need to know how to order these trinomials, because they're giving them to me out of order. If I can understand how the order works, it should be easy.

- hedgepig

i already gave you the answer to this - you order them from highest degree to lowest.

- anonymous

The problem is, I genuinely do not know what the highest degree is. I don't.

- hedgepig

note that for expression x^4 y^2 z^7 the degree is 4+2+7=13

- hedgepig

Yes. That is, again, why i told you to identify the degree of each term.

- hedgepig

6x^3
23x^5
x^3
8x
9
-5x^2

- anonymous

9-6a-24a^2

- anonymous

So, 24a^2 is first

- hedgepig

of what degree? and you forgot a minus sign.

- anonymous

The other two don't have a degree, though.

- anonymous

Yes, my bad.

- hedgepig

I don't understand why you keep ignoring me.
Identify the degree of each term
6x^3
23x^5
x^3
8x
9
-5x^2

- anonymous

The degree is the 2.

- anonymous

I'm sorry, I'm not trying to ignore you.

- hedgepig

The degree isn't the 2 in that case. The degree IS 2.

- hedgepig

|dw:1442986372632:dw|

- anonymous

1. 3, 2. 5, 3.3, 4. IDK, 5. IDK 6. 2

- hedgepig

5 is 1, and 6 is 0.

- hedgepig

x^1=x
x^0=1

- anonymous

Okay.

- hedgepig

it might be hard (I know I had a bit of trouble with exponents in the beginning) but you really need to know how to identify and visualize what's going on here.

- anonymous

9-6a-24a^2
Is -24a^2-6a+9

- hedgepig

for term
ax^n for constant a and n, the coefficient is equal to a and the degree is equal to n
for term bx^n y^m the coefficient is b and the degree is equal to n+m

- hedgepig

yes... but, what I don't understand is how you are able to factor without knowing how to order polynomial terms

- anonymous

Perhaps that was the only reason I was having confusion. I would like to work on the problem a bit, see if I get it right, and you can confirm.

- anonymous

Well, I guess we'll see if I know how to factor. I think I do, though. :)

- hedgepig

the answer is aways above if you wish to check it.

- anonymous

Yes, but I do want to do it myself.

- hedgepig

my suggestion to you, if you are so dedicated and inclined, and if you wish to have future success in mathematics, is to review prealgebra

- hedgepig

there are plenty of resources available online for free to do so...

- anonymous

Anyways, if I did this correctly, I have to think of two factors that add to -6 and when multiplied, are -216. Is this what I am supposed to do?

- hedgepig

where are you getting -216

- anonymous

24 x 9

- anonymous

-24 x 9

- anonymous

Okay, so... am I to square the first and last number?

- hedgepig

actually, let me think, I have never learned how to factor that way.

- anonymous

Well, 24 isn't a perfect square. So, honestly, I have no idea what to do from here.

- hedgepig

nope, that doesn't look right at all.

- hedgepig

I was walking you through the problem step by step, but I must've gone too quickly.

- hedgepig

Do you understand the purpose of factoring trinomials, and what the inverse operation of that is?

- anonymous

My understanding is that the purpose of factoring is to simplify.

- anonymous

The inverse operation, I would assume how we're all familiar with foil, well the inverse operation would be the system to get a trinomial in into the state of foil. How it's implemented, though, I'm not sure.

- hedgepig

well why don't we look at it.
let's foil
(ax+b)(cx+d)
what do you get?

- anonymous

Give me a second.

- anonymous

Ok.

- anonymous

I got acx^2+axd+bcx+bd

- hedgepig

ok.
that's equal to (ac)x^2+(ad+bc)x+(bd) right?

- hedgepig

remember our expression
-24a^2 -6a+9
?
here is what factoring really is.
ac=-24
-6=ad+bc
9=bd
Solve the system for a, b, c, and d

- anonymous

I don't understand the question.

- hedgepig

Note that, once we have the solution, we know that the factored form of the expression is simply
(ax+b)(cx+d)

- anonymous

You want me to tell you which each letter signifies individually? I don't know how to do that.

- hedgepig

no. I just gave you a system of equations.

- hedgepig

ac=-24
-6=ad+bc
9=bd

- anonymous

Yeah, I don't understand what that is asking.

- hedgepig

find values of a, b, c, and d such that each of the three equations holds true

- anonymous

Oh, okay.

- anonymous

Will do. :-)

- anonymous

Give me a minute, please. Thank you.

- hedgepig

this is how I got the system by the way:
(ac)x^2+(ad+bc)x+(bd)
-24a^2 -6a+9
the variable a is redudant but for the second equation take x=a to get
(ac)x^2+(ad+bc)x+(bd)
-24x^2 -6x+9
well,
(ac)x^2+(ad+bc)x+(bd)
(-24)x^2+(-6)x+(9)
thus, (ac)=(-24); (ad+bc)=(-6);(bd)=(9)

- hedgepig

by the way, here is a hint: you should not try to solve the system of equations using algebra. Use trial and error.

- anonymous

Okay. :-) I knew that.

- anonymous

A = -6

- anonymous

C = 4

- anonymous

B = 3

- anonymous

D = 3

- hedgepig

and what will our factored expression be?

- anonymous

I don't know.

- hedgepig

what did I tell you to foil?

- anonymous

Sorry, I do get stressed when I work on math. My brain usually isn't at its peak at this point.

- hedgepig

you are doing just fine - I am not approaching this problem in the standard way that most people learn how to factor... I choose not to do so because you will learn the standard way from everyone else...

- anonymous

You wanted me to foil (ax+b)(cx+d)

- hedgepig

ok... so simply resubstitute your values for a, b, c, and d, no?

- hedgepig

What do you get?

- anonymous

Give me a moment.

- anonymous

Okay, I don't understand. I didn't figure out the meaning for an X.

- anonymous

I do appreciate you trying to be innovative in your explanation, but perhaps the standardized explanation is so for good reason?

- hedgepig

mmm this is a key point that most people struggle with algebra.
We first introduce letters that represent numbers in algebra.
Well, we often tell people to evaluate expressions by assigning them values for particular letters.
Here's the thing though - there are basically two kinds of symbols: symbols that represent CONSTANT numbers and symbols that represent UNKNOWN numbers. You might wonder, why is it that we need symbols for CONSTANT numbers? Consider the polynomial of degree two
ax^2+bx+c
There are infinitely many polynomials of this form. here are a few.
x^2+3x-3
10x^2-12x+5
but what about x? we don't actually have any fixed value for x, so we leave it as is, despite knowing what a, b, and c are.
In this case, i have given you an expression with a, b, c, d, and x and we have found values for a, b, c, d... but not x... why is it that we do not know what to do for x?
Simple. Let's look at the original problem.
9-6a-24a^2
note that halfway through I turned the a's to x's. No matter. From the start, we defined a (x right now) as a symbol that varied. There is no way to find it.

- hedgepig

The standardized explanation is why many of my peers struggled deeply with calculus - not because they could not follow the rules of calculus, but because they could not CREATE expressions from understanding algebraic motivations.

- anonymous

Okay.

- hedgepig

I am not saying not to look at the standardized explanation - I have a pretty good site I will refer you to after this, but we are almost at the end of this exercise.

- anonymous

Yeah, by all means continue. You know what you're doing. :-)

- anonymous

-24^2-6a+9 I just don't know how to factor it.

- hedgepig

for
(ax+b)(cx+d)
the substitution
a = -6
c = 4
b = 3
d = 3
yields? do not worry about foiling the result

- anonymous

Okay. I will do that.

- anonymous

(-6x+3)(4x+3)

- hedgepig

You basically just factored the expression. Now, there is one last step we must perform to call out expression factored. We can actually factor 3 out from the first parenthetical

- hedgepig

*-3 actually

- anonymous

So it would be (-2x+3)(2x+3)-3

- hedgepig

yes.. but when you have a factored expression, you actually put any constants to the very front. so your -3 belongs in the front of everything.

- hedgepig

and I think you meant, for the second parenthetical, 4x+3

- anonymous

Okay, so how do I do with a trinomial?

- hedgepig

what do you mean?

- anonymous

The original problem

- anonymous

I still honestly don't know what to do.

- anonymous

I just became you're first fan, by the way.

- hedgepig

hahaha I am on an alt, but, yes, first fan on this account :)

- hedgepig

(-2x+3)(2x+3)-3
move the negative three to the other side

- anonymous

-3(-2x+3)(2x+3)

- hedgepig

erm, 4x+3 sorry

- anonymous

?

- hedgepig

sorry we are really close to the answer and I don't want to steal it from you by typing it ahaha

- hedgepig

the second parenthetical should be 4x+3... you miscopied it before and I did the same

- hedgepig

(-6x+3)(4x+3)
->(2x-3)(4x+3)*-3

- anonymous

-3(2x-3)(4x+3)

- anonymous

:-)

- anonymous

Did I do it correctly?

- hedgepig

yeah... you just went about my roundabout method of factoring the expression (and you got it right! though those x's are supposed to be a's :P http://www.wolframalpha.com/input/?i=factor+9-6a-24a%5E2+ ). if you wish, you can remultiply to ensure that it is correct.
but, as I promised, here are some links to help you with factoring
this is the homepage of the site I learned algebra from
http://www.purplemath.com/modules/index.htm
http://www.purplemath.com/modules/factquad.htm
here is the famous khan academy
https://www.khanacademy.org/math/algebra/introduction-to-polynomials-and-factorization/factoring-polynomials-2-quadratic-forms/v/factoring-polynomials-1

- hedgepig

pertaining DIRECTLY to what we just did is http://www.purplemath.com/modules/factquad2.htm

- anonymous

Okay, but I still don't know how to do the original problem, it's a trinomial.

- hedgepig

all of those links deal with trinomials. quadratic refers to the fact that they have maximum degree 2.

- anonymous

You know, I sometimes do need a little bit of a different approach.

- anonymous

Could you tell me what is the step you would take in solving the original problem now that we know the proper order?

- hedgepig

here is a computer generated solution manual

##### 1 Attachment

- hedgepig

note that this is also called factoring by grouping

- anonymous

Okay, I saw the next step. I didn't see the answers. I'll like to see if I got it right. :-)

- anonymous

So, let's see if I get this! :-)

- hedgepig

mmhmm - I think this site has the lesson you are looking for fyi
http://www.purplemath.com/modules/factquad2.htm

- anonymous

Working on it. Hope you stay. :-) Thank you so much for your time.

- anonymous

Okay, did -3 HAVE to be factored out? My natural proclivity was to factor 3.

- hedgepig

haha no problem -thanks for bearing with me even when it hasn't been apparent where I've been taking the discussion. unfortunately, it's around 2:50AM and I have a chemistry class tomorrow at 9AM, so I should probably head off to bed.
As for -3, it is standard to make the leading coefficient (the number in front of the a^2 in this case) to be positive, because otherwise all the other signs have to be switched while you try to factor.

- hedgepig

the point of using -3 was to make it so the number in front of a^2 was positive

- anonymous

Okay. Are you in college? What year are you?

- hedgepig

I'm a freshman in college.

- anonymous

Ah, okay. Yeah, I better let you sleep. It is important.

- hedgepig

best of luck to you - feel free to respond to those post if you have anymore questions, though I would actually suggest you open a new question with the same problem and have some other persons look at it - they will give a much more straightfoward way of explaining how to factor.

- anonymous

What's your major?

- hedgepig

undeclared engineering - are you in college?

- anonymous

Nope. Junior in high school.This is Algebra 2

- anonymous

Does my work seem college level? :p

- hedgepig

No, but you would be surprised as to the number of students who take "college algebra" which is merely junior high school algebra in college.
Your work does not seem college level, but your vocabulary does seem so.

- anonymous

I'm a bit confused as to what to do with +6 and -4

- anonymous

Could you illustrate that?

- hedgepig

hmm one moment

- anonymous

I'm sorry, I hate to take up your sleep, but hopefully, once I'm certain how to do one question, I'll know how to do them all.

- hedgepig

so, you have the expression (8a^2+2a-3)
well, let's look at (ax+b)(cx+d)=(ac)x+(cb+ad)x+(bd)
note that this is saying
we search for two numbers that have a product of 8. (ac=8)
we search for two numbers that have a product of -3 (bd=-3)
we seek to make ad+bc equal to 2.
this is where the 6 and -4 come in.
6-4=2, which means ad=6 and bc=-4, or ad=-4 and bc=6.

- hedgepig

why did we pick 6 and -4? well, it was a natural choice because if we look at the factors of 8, we have 2 and 4 and 1 and 8. if we look at the factors of -3, we have 1 and -3 or -1 and 3

- anonymous

Yes, I know how to get the factors, but I don't know what to do with them.

- hedgepig

well, there are a lot of choices for the factors of 8, so we will worry about the factors of three instead.

- hedgepig

thus we wind up with something like
(?a+3)(?a-1)

- hedgepig

well, we will wind up with 3?a-?a such that we get 2a when we pick two question marks. note the question marks have to multiply to 8. because ?a*?a=8a^2

- anonymous

My guess, would be (8a^2+6) (4-3) (1) 4-3 is 1, which we can cross off, so -3(8a^2+6)

- hedgepig

hold on... the answer -3(8a^2+6) is not what we originally looked at. if you simplify that, you get somethign different fromt he original expression.

- anonymous

I don't know, going to need your explanation skills 'cause I'm lost.

- anonymous

Says you've been typing for a while.

- hedgepig

let's start over.
we just factored out that -3
-3(8a^2+2a-3)
so let's worry instead about factoring 8a^2+2a-3.
Well, all this is asking you to do is to rewrite 8a^2+2a-3 in the form
(ra+t)(qa+s)
well, factoring is kinda one of those things wher eyou guess and check.
we can foil out our expression to visualize it better though.
we'll get
(rq)a^2+(tq+rs)a+ts
compare it with 8a^2+2a-3 in
note that 8 HAS to equal rq
and -3 HAS to equal ts
and 2 HAS to equal tq+rs.
Why? because our expression is in the most simplified form possible. we know that all the like terms have been combined, so we can match the coefficients safely.
it then boils down to solving the system.
to do so, let's compare the pairs rq and ts and see which one has fewer potential factors. the factors of 8 are 1,2,4,8 (or -1,-2,-4,-8) while the factors of 3 are (-1,1,-3,3) so let's start with picking arbitrary factors of three for our values ts.
so, let's say t=3, and s=-1
then, we have (ra+t)(qa+s) -> (ra+3)(qa-1)=8a^2+2a-3
well we know they already multiply to -3 because we chose them to.
however, we're gonna have to make additional constraints to this problem if we want to get anywhere. let's pick values for r and q.
remember rq=8, so let's just do r=2 and q=4
then (2a+3)(4a-1)... well we are really just trialing and erroring for the value tq+rs.
in this case we get tq=12 and rs=-2
tq+rs=10. We're supposed to get 2.
so, let's think... how can we get 2? well, it just so happens that if tq=6 and rs=-4, we'll get 2. That intuition comes with expreience, but that is the origin of the 6 and -4.
knowing that, it follows that because ts=-3, t=3, and q=2. whereas, s=1, and r=-4 that is the solution in this case, which you deducted earlier actually.

- anonymous

Okay, I know how to figure out the factors. That's easy to me.

- hedgepig

then what isn't?

- anonymous

6 and -4. I struggle with what to do with the factors, though.

- anonymous

That is my struggle.

- anonymous

I'm fine with the guess work.

- anonymous

8a^2+2a-3 and we have 6 and -3. I don't know what to do from here.

- hedgepig

just resubstitute. again, with the expression
(ax+b)(cx+d) you will get
acx^2+(ad+bc)x+bd...

- hedgepig

your 6 and your -4 are, respecitvely, your ad and your bc in this instance.

- anonymous

So (8a^2+6) and (-4-3). Is that right?

- hedgepig

... no - foil the result and you will quickly learn that that does not yield what you need it to
I am not sure i can do a good job explaining this... I would advise you check out some videos on khan academy.
I am basically out of time but not done explaning, so I will give you this video which uses another method (factoring by grouping) and hope it will help. It's 5 minutes...
https://www.khanacademy.org/math/algebra/introduction-to-polynomials-and-factorization/factoring-polynomials-2-quadratic-forms/v/factoring-trinomials-by-grouping-6
if you have time, I'll be online very shortly tomorrow morning.

- hedgepig

(in around 4 hours)

- hedgepig

bonne nuit

- anonymous

Good night!

- anonymous

Funny you say that, as I am studying French.

- hedgepig

If I could have I would've asked you to show your work that got you to that result, but unfornuately I really am out of time - no point in me showing up to chem class if I can't even think

- anonymous

Fair enough. Thank you so much for staying as long as you did; I'm sure it was frustrating dealing with me.

- hedgepig

I "studied" french - all i remember are the very, very basics now, unfortunately.
There was no frustration dealing with you by the way - it is incomparable how often I get frustrated with entitled kids who come on this site demanding answers with neither appreciation nor patience. You have a good degree of both, and you are one of the rare cases where the teacher leaves before the student.
I will admit that I am somewhat frustrated though - not at you for not understanding the concept, but at myself for not being able to articulate that which i supposedly find so straightfoward and easy to do.
Farewell!

- anonymous

See you!

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