Identies, algebraic expressions

Hi everyone,

I've just joined TSR, and this is my first post - so please excuse me if I am posting to the wrong area.

I'm currently in the process of teaching myself A-level maths. I've bought myself an AQA A-Level student book which I am working through, and am slightly stuck with a question:

"X^3 - 4X^2 - 3X + 18 = ( X + a ) ( X - b )^2 for all X.

Find the values of a and b."

My question is 2 part:
1) What does the question mean by "for all X"?
2) How would I go about solving this?

The only way I've managed to figure it out is by saying,

Well, 18 must be the product of (a)(b^2).

If that is the case, there are only 2 combinations of A and B that can solve this expression. Either A=18, B=1, or A=2, B=3. I can then simply plug these numbers back into the original expression to find out which combination works.

Obviously however, this isn't a very robust method. With some expressions I'm sure there could be 10+ combinations, and working through each one is impracticle.

So how do I solve this?

Thanks!

Hi, firstly welcome to TSR! Please post maths questions in the Maths form - I have moved this thread for you

1) This is the difference between an identity and an equation. An equation may be true for 1 or more values of x but an identity is true for all values of x.

E.g. $x+3=4$

This is an equation since it isn't true for all values of x like x = 2 is false. The only solution is x = 1.

But $2(x+3) \equiv 2x + 6$

This is true for all values of x and here we use an identity symbol $\equiv$. An identity means that the left-hand-side is the same as the right-hand-side.


2) You don't just have to focus on the constant term. Try expanding $(x+a)(x-b)^2$ fully and then compare coefficents for every term. Post your working if you're unsure.

With identities another technique you can use is plugging values into x into both sides of the identity. E.g. plug x = 1 into both sides and you will get an equation with a and b. Plus x = 2 and you will get a second equation. You can do this because it's an identity so must be true whatever x is.

Hi Notnek,

Thanks for the reply.

1) I think I understand now. I knew identiies used \equiv instead of =, but the question actually uses =. So I a gather "for all of X" is essentially how they have chosen to (for lack of a better word) identify the expression as an "identity"?

2) I tried expanding the brackets fully, and got: x^3-4x^2-3X+18=x^3-2bx^2+xb^2+ax^2-2bax+ab^2

To me, I can remove x^3 from both sides. Then -4x^2=-2bx^2 (which I take to meaning b=-2), which leaves me with -3x+18=xb^2+ax^2-2bax+ab^

I don't really know where to go from here?

I've tried plugging b=-2 back into the original expression, but that doesnt work (and is definitely wrong as b=3).

Thanks

Yes that's right. An exam question would probably use an $\equiv$ symbol but sometimes they are left out in textbooks.


The next step after expanding is to group the terms. So on the left-hand-side there is an x^3 term, and x^2 term, an x term and a constant term which means that if you are to compare coefficients, the right-hand-side must have the same polynomial structure.

For example, there are two x^2 terms on the right-hand-side so you need to group them by factorising:

-2bx^2 + ax^2 = (a - 2b)x^2

Try doing this for all terms and then compare coefficents. Again post your working if you get stuck.

Hi Notnek,

OK, so grouping the terms has left me (providing I've done it correctly) with x^3 + (a - 2b)x^2 - (2ba)x + (x + a)b^2 = x^3 - 4x^2 - 3x + 18.

From here, comparing the coefficients gives me:

-4 = a - 2b
-3 = -2ba
18 = (x + a)b^2

Which still leaves me stumped

Hi Notnek,

OK, so grouping the terms has left me (providing I've done it correctly) with x^3 + (a - 2b)x^2 - (2ba)x + (x + a)b^2 = x^3 - 4x^2 - 3x + 18.

From here, comparing the coefficients gives me:

-4 = a - 2b
-3 = -2ba
18 = (x + a)b^2

Which still leaves me stumped

Actually if you start by assuming a and b are integers then should be able to find a and b easily (similar to your reasoning in your first post) without doing any algebra manipulation.

But ths wouldn't work if a and b weren't integers.

Something else potentially useful here is that if

p(x) = (x+a)(x-b)^2, then

dp/dx = (x-b)^2 + 2 (x+a)(x-b), which is 0 when x = b.

In other words, if you differentiate, and find the roots of the quadratic, one of the roots must equal b. (And you can then easily find a from the other root).

I don't know if at A-level the "assume the answers will be integers" is actually backed up (albeit only at the "there's a theorem you don't need to know the details of" level), or it's just a general rule-of-thumb.

Are you not told that a and b are integers? This seems like the kind of question where you are.

Assuming you are not, firstly you made a mistake when grouping your x terms - you've grouped b^2 instead. There should only be 1 term that has an x in it.

You should end up with 3 equations and you can choose two of them to solve simultaenously. For this you'll need to make either a or b the subject of an equation and plug that into another equation.

Hi Notnek,

No, there is no information given regarding integers. The entire question is what I posted originally.

I see where I've gone wrong, and grouped to give:

x^3 + (a - 2b)x^2 - (2ba - b^2)x + ab^2 = x^3 - 4x^2 - 3x + 18

Essentially giving: 4 = (a - 2b), 3 = -(2ba - b^2), 18 = ab^2

I'm struggling to see how I can solve any of these equations. Am I being completely stupid?

The signs in your equations are slightly off and it should be -4 = (a - 2b) and -3 = -(2ba - b^2). Can you see why?

From here well you could assume they're integers (they have to be as mentioned above but you're not required to know this) and then you can find integer solutions to 18 = ab^2 like you did in your first post and check which pair satisfies -4 = (a - 2b).

Or you could make a the subject of "-4 = (a - 2b)" and then substitute for a in "18 = ab^2". This will give you a quadratic in b which you can solve. You should have solved non-linear simultaneous equations like this before? Although I don't know when you did GCSE or if you've got to that part in A Level yet.

Hi Notnek

I did well in my GCSEs (A), but haven't done them for 7-8 years, and have only just started A-Level maths, so haven't really covered anything yet.

Can you maybe suggest topics I should cover before moving forward with A-Level?

It's been so many years, and I've not had to use maths at all as I work in IT, so I'm very rusty.

Thanks

You've come to the right place. This forum is perfect for people self-teaching to post their questions if they don't have a teacher/tutor.

You did well in GCSE so you don't need to revisit everything but I think it's worth going through all the algebra topics to make sure you remember how to do it all. You'll be doing loads of algebra at A Level and a lot of GCSE topics will be assumed knowledge. Maybe try a GCSE exam paper to see how much you've forgotten.

A GCSE textbook would be good to have if you don't have one already and a topic list like my one could be useful. Try going through some of the algebra topics in my list (click on the links) to see how much you remember. A lot of the other topics will be useful like trignometry and vectors. Are you aware of sites that have videos explaining topics? These are extremely useful if you don't have a teacher. Also, do you have any timescale for when you want to finish learning A Level?

Please let me know if you have other questions and feel free to post maths questions in this forum whenever you're unsure about something or need help

Hi Notnek,

Thanks so much. You've been really helpful.

I'll go back over all the algebra topics as you've suggested, and look at some mock exams to highlight any other areas I need to brush up on. I know BBC bitesize has some mock exams which may be useful. Not sure of any other sites other than bitesize and TSR. Recently I've been using Khan Academy to study Quadratics & Polynomials, and there is alot of other good content on there. Do you have any suggestions?

I've actually been looking for a career change for over a year now, but haven't been sure on where I want to move to. A few weeks ago I came across mechanical engineering, which I think is what I want to do goign forward. I want to do a Masters in Mechanical Engineering at the Open University (which has no formal requirements). So there isn't really a deadline on the study, although I would like to get math & physics done in a year.

I know it's very optimistic, but I am very committed and have been doing around 4 hours a day outside of working full time.

Lots of people have used this site in the past when self-teaching and most have done well although generally these are students who are self-teaching before they learn the material in school so you'd expect them to do well.

For mature students who haven't done GCSE for a while, the results have been a bit of a mixed bag. One of the biggest problems is that they lack the understanding due to not having a teacher. This is why videos and asking questions on e.g. this site is so important. You must make sure that you are never doing textbook questions by just repeating the examples given without actually understanding what's going on. The more questions you ask yourself the better. I'm not trying to scare you - just warning you that self-teaching A Level maths is tough and requires a lot of work. It sounds like you're up for the challenge though!

The best site for someone in your position is ExamSolutions (scroll down to the new specification). He has videos on every topic and they're really good. Plus he takes you through loads of exam questions. If I was in your position, for every topic I learnt in my textbook I would also spend watching his videos (usually < 10 mins) that explain the topic. But I think you should look at GCSE topics first before starting this, as I said before.

I assume you have textbooks - are they designed for the new spec? Which exam board are you using?

I think it's worth mentioning that while you're not explicitly told that the roots are integers, in my experience of A Level maths you're sorta expected to guess a couple of integer roots and check with remainder theorem. I'd say at least try integers between -3 and 3 inclusive. Then you can factorise further when you've got a root (by long division or whatever other method).