### Perkalian Bentuk Aljabar Soal dan Pembahasan Penting (Kelas 7 SMP)

Kali ini kita belajar mengalikan operasi bentuk aljabar dengan mengikuti hukum distributif.

Contoh 1

Kalikan *x*^{3}(*x*^{4} + 5*a*)

Jawab

Kita perluas dengan hukum distributif

x^{3}(x^{4}+ 5a)=

x^{7}+ 5ax^{3}

Kita sudah tidak dapat menyederhanakan lagi.

## Contoh 2

Kalikan (*x* + 5)(*a* − 6)

Jawab

Uraikan hasilnya sebagai berikut :

(

x+ 5)(a− 6)=

x(a− 6) + 5(a− 6)=

ax− 6x+ 5a− 30

Sudah tidak dapat disederhanakan kembali. Selesai sampai di situ.

## Contoh 3

Kalikan (2*x* + 3)(*x*^{2} − *x* − 5)

Jawab

Kita kalikan dengan memecah

(2*x* + 3)(*x*^{2}− *x* − 5)

= (2*x*)(*x*^{2} − *x* − 5) + (3)(*x*^{2} − *x* − 5)

= (2*x*^{3} − 2*x*^{2} − 10*x*) + (3*x*^{2} − 3*x* − 15)

= 2*x*^{3} + *x*^{2} − 13*x* − 15

Kita jumlahkan dengan melepas kurung.

−2

x^{2}+ 3x^{2}=x^{2}

dan

−10

x− 3x= −13x

## Contoh 4

Kalikan $\displaystyle{\left({x}-{3}\right)}^{2}$

Jawab

(*x* − 3)^{2}

= (*x* − 3)(*x* − 3)

= *x*(*x* − 3) − 3(*x* − 3)

= *x*^{2} − 3*x* − 3*x* + 9

= *x*^{2} − 6*x* + 9

## Catatan penting:

$\displaystyle{\left({x}-{3}\right)}^{2}$

tidak sama dengan

$\displaystyle{x}^{2}-{9}$

Contoh 5

Sedrerhanakan 5*x*[−4 + 10(*x* − *y*)] + 7*x*

Jawab

5*x*[−4 + 10(*x* − *y*)] + 7*x*

= 5*x*[−4 + 10*x* − 10*y*] + 7*x*

= −20*x* + 50*x*^{2} − 50*xy* + 7*x*

= −13*x* + 50*x*^{2} − 50*xy*

## Contoh 6

Sederhanakan (*p* − 1)(*c* − 1) + 2

Jawab

*p*(*c* − 1) − 1(*c* − 1) + 2

= *pc* − *p* − *c* + 1 + 2

= *pc* − *p* − *c* + 3

### How to Find an Effective Way to Think of a Solution when Trying to Solve Them

Others have recommended practice, practice, practice. I recommend changing your level of understanding for the subject of math so you can “find an effective way to think of a solution when trying to solve them.”

Do to this, I recommend you find a tutor, form a study group, locate a volunteer minister or other person or persons that can help you with just one action. Even though it is simple, I don’t think you will be able to do this action all by yourself.

Once you have found this person or formed this study group, you need to do this little program.

First you need to learn this definition for understanding:

“To have a clear and true idea or notion of something, or full and exact knowledge of something.” This is learned as a skill in itself.

Second, locate one word you do not understand, that you have not understood, that you have gone past, in the subject of Math. Define that one word. You define it in a special way. First you locate the most appropriate meaning in a good math glossary, or text book, or dictionary. You clarify that one definition first, read some sample sentences, use that meaning in sentences that connect that meaning to what you will be doing in life. Make up some examples of how that concept can be used in life. Demo how you could apply the concept to things in life. You demo this several times till you can think of an effective way to apply it to things in your life, more or less instantly. This might take 10 or more sentences, 5 or more examples and 3 or more demos. Then you define the other common definitions. You just have use each of these in sentences until you make it your own. Then learn the etymology, and any usage notes, or idioms. You do this technique with any word you define in this program.

Third, you locate an earlier word you did not understand in the subject of math. You define that. You keep locating earlier words in the subject of math. When you and your tutor, or your study group locate and define the earliest one I would expect the following: Your ability to learn the subject of math should increase dramatically, all of your understanding for the subject should return, your willingness to learn the subject should go way up, your capacity to understand and your willingness to understand the subject should go way up, and your ability to “find an effective way to think of a solution when trying to solve them.” should be way better. If this does not happen. Let me know. I would not expect this to be more 10 words. But no matter how many words you find, it will worth it.

After you find and define that first word in the subject you then restudy your materials from just before that time in your life, forward from there, up to where you are now in your studies now. Do not go past a word you do not understand fully. Define each word that you do not fully understand in this way.

This all should not take longer than 8 hours. If it does let me know.

Now is the time to practice. But with one more special thing. Each time you find, if you do, a problem that seems too difficult for you at this new level, you stop. Look it over and with your study group, find the word that is blocking your understanding of that problem. in this case you might not have covered the knowledge you need to solve the problem. In this case you need someone that can solve that sort of problem to point out what math material you need to master to do that sort of problem. Keep locating these progressively harder problems and locating the people that do know how to solve them.

Good luck on your adventure. Let me know from time to time how things are going if you do decide to use my program as written.

Writer : Leon Hulett from Quora

### Best Ways to Get a Problem Solving Skills

I am no math genius, but over time I think I have picked up some approaches that generally work to help improve problem solving, and in particular, applying math to problem solving. I wasn't entirely sure if you were talking about focusing on a particular area. My answer is general. I hope it is useful.

Practice

In the end it comes down to practice. But how one practice's can make a big difference.

When you solve a problem you are working on, congratulations ... but you are not done with it. Look at it in other ways.

Change Constraints

One way I try to practice is by blowing things up to extremes. What if this was zero, what if that blew up to infinity, what if this was an infinite sequence. Things like that.

Here is a smattering of substitutions that I sometimes think of. This is totally random. You might imagine starting each of these bullets with the phrase "What if ...", followed by, "... how would that affect the problem".

- non-linear instead of linear
- stochastic versus not
- n dimensional instead of m dimensional
- statistical approach versus closed form
- not differentiable instead of differentiable

and so on.

Invent Your Own Problems

The "technique" I described above leads into this idea well. Invent a different problem by changing an existing one.

Teach

Teach things to others. Many argue that is the best way to learn something.

### What do You Do if You will Get Good at Mathematics for Problem Solving

This is a general question, but here are a few hints, which may be directed at your parents if you're young enough to have parents guiding your life still

I'm not going to say much about mathematics itself since there's a lot of it out there, and plenty of books and I don't have enough space.

Step 1. Get yourself 3 copies of A Mathematician's Lament by Paul Lockhart.

The book is available freely by download from various places and also as an Amazon Kindle book.

This book will not solve all your problems, but might get you asking some of the right questions. The three copies are for yourself, your parent (if there is one) and whoever is currently teaching you math, and for you all to compare notes after reading it.

Dr Lockhart has pet peeves and great ideas, and it's up to you to separate them according to your own abilities and inclinations.

2. To be good at mathematics, it needs to ascend a notch from a subject at school to a particle of your being, just like Sunday School (sorry if you're not Christian, just an example) is intended to teach you to be a good Christian all the time, not just on Sunday mornings.

3. To achieve this, and make a truly functional difference in your mathematical achievements, I suggest, on behalf of all the gifted mathematics students of my acquaintance (and I've known quite a few) that you need to upgrade your foundation skills, starting with arithmetic. This could seem burdensome, so it's best to couple it with a game, such as being able to estimate the value of a shopping cart without the use of a calculator. This is a valuable life skill, never mind how far you go in math, so how about going in that direction

4. Another book that may help you integrate math into everyday like is Mathematics for the Million (various editions)

5. If you're past the high school level, you might consider the book Companion to Concrete Mathematics by my old friend and teacher Z. A. Melzak. This book in particular (available as a Dover reprint) is not a textbook full of tedious things for you to do. It's a treasure-trove of difficult mathematical problems, still brought within the grasp of an undergraduate student. They are for your amusement as much as anything.

6. That's where we have to go --- making math fun. The other persons mentioned above can take inspiration from Mathematician's Lament to formulate an action plan. The plan should extend through summer breaks and through your entire effort to become more mathematical.

7. You should pay attention to related matters like Physics and Economics because they are fruitful contemporary areas of application of math. Economics in particular is neglected in the old textbooks, but many modern business textbooks have applications. See The Mathematics Behind Wall Street for an example -- only the most basic math is needed to grasp that book.

Writer : Bruce Balden from Quora

### How to Increase Problem-Solving Skills in Maths and Physics?

I'm with Jung, You need to be really interested in the subject, or equivalently be interested in something where maths and physics can be applied, like engineering, robotics, computer games, flying rockets, model airplanes.

Take my son for example, at grade 8 he got into writing computer games, so you need really good trig to this as well as partial differentials so in primary school he ended up explaining coordinate rotation to the teachers. He also needed an understanding of physics to get his objects bouncing off things, and flying through the air. Needless to say he taught himself computer programming too.

If you don't know how to work a chisel and saw, and it's really difficult and demoralising for you , then take up carpentry or sculpture , once the subject becomes a "tool of trade" then it is intensely motivating, learning how to skilfully use your tools of trade is no longer a chore as it has a purpose.

So your suggestion of "solving a lot of problems" is half the answer, you really need to find an activity that requires you to solve problems, this generates the hunger for knowledge that you seem to be lacking.

Having a study group , or a common interest club helps, you can learn a lot about something by explaining it to your peers, and the competition instinct motivates you to do better than your peers.

When I started university 43yrs ago, there were two sorts of engineers

(a) guys like me who had been messing around with electronics as a kid, for us , all the subjects we learnt explained all the behaviour we had already seen, and we had experimented with, and we just sucked it up like sponges.

(b) guys with really good GPA's but no clue what they wanted, they had to study hard and it all seemed pointless, all the info kept falling out of their heads like a leaky sieve. They were also useless at all the prac work too.

Some of the (B) students might be 20IQ points smarter than us, but us (A) guys could help them understand because we understood the application of the subject.

### How to Improve your Mathematical Problem Solving Skills?

I generally think solving a maths problem is one part experience, one part skill, and one part luck. So let's take these one-by-one.

Experience:

This is the most obvious one and other people will no doubt mention it so I won't belabour the point. The more problems you solve, the better you will get. But there are a few caveats:

- Don't just solve problems you can easily do already. Solving a million quadratic equations is not going to improve your analysis. Splitting a million expressions into partial fractions will not improve your group theory. Using the cosine rule a million times will not improve your calculus. I’m sure you get the point. Getting the groundwork is important, and it's crucial to be able to do the basics before you put them in a problem solving context, but you can't just do this.
- On the other hand, there’s no use attempting problems you can’t even begin to understand. If you’re very new to calculus, looking at second order differential equations isn’t going to help you differentiate polynomials. If you don’t have a clue how to do any of the questions on an exam, you should move to an easier one. In my experience, you learn the most when you can do 60–70% of the problems you attempt. That way, there is plenty of opportunity to learn new techniques, but you won’t get discouraged, and often finding the answer yourself helps the relevant technique stick in your mind. Everyone remembers some difficult problem they finally cracked, but almost no one remembers a difficult problem they looked up.
- Persevere with the questions you’re trying. If you’ve only been working for a few minutes, there’s absolutely no use looking to the answers. For me personally, I keep grappling with a problem until I reach a stage where in the last 20–30 minutes I’ve made absolutely no progress. At this point, I briefly glance at the solution to get some kind of hint, then keep working myself. The longer you work on a problem, the more you benefit from knowing its solution.
- You must actually look at the solutions. You can still derive some benefit from answering questions and ignoring those you get wrong, but you’ll benefit so much more if you read the solutions afterwards (even if you already solved it!). And when I say read, I mean take the time to carefully work through the solution given, making sure you understand every step, and making sure you know where you went wrong.

So the more experience you have, the more you will have trained yourself to think critically about problems. But it goes deeper than that. Eventually you’ll start to recognise problems, or at least see similarities between new problems and ones you’ve already solved. You’ll develop much better intuition and have a larger arsenal of attack.

Skill:

A lot of people think of skill as something you’re born with — you either have it or you don’t. To a certain extent, I agree, some people are more mathematically inclined than others, but at the same time any given person can improve enormously with the right training.

A large part of skill comes from experience. The more maths you do, the quicker you’ll be, and you’ll be more accurate too. But there’s an equally significant portion which comes from the right mindset. There are entire books dedicated to the subject (I’m looking at you, How to Solve It) but briefly:

- You need to be able to examine a problem and reduce it into pure information. If you’re doing a set of structured questions, there will very rarely be irrelevant information given to you. So look at what the question is telling you, think about what it means, and what it implies. Think how the various pieces of information link together and keep repeating this process to make sure you bleed the information for everything it has. Of course, if you’re doing original mathematical research, the state of play is quite different. You have no idea what information will be helpful and what will be redundant. However, I find the best tactic is to keep a separate sheet of paper where you jot down any intermediary results you’ve proven, regardless of whether you think they’re helpful, and keep looking at it for inspiration.
- You should be able to break a problem down into smaller problems. If you can’t solve something, think of a similar, but simpler, problem and solve that. Maybe you could solve the case
n=2 $n=2$, or solve it whenn $n$ is prime, or solve it when the quadrilateral is a square, or one of the variables is fixed, or whatever. Try to relate the techniques used in the easier cases (or indeed the result itself) to the case in hand. - Finally, you should be able to check your answer. I don’t just mean going through the algebra, though that’s certainly useful, but ask yourself whether the answer makes sense contextually. If you’ve calculated that a person walks at 100 mph, or they weigh a million tonnes, then you’re almost certainly wrong. But those are pretty obvious examples; often it can be more subtle. Is your equation symmetrical about the variables and should it be? What happens as a value goes to infinity, or zero? Has your working created redundant solutions?

Luck:

This is the part over which you have the least control, but of course experience still plays an important role here. Often what we call luck is actually just a super refined intuition. There is a common expression that says “the more I practice the luckier I get” and I think it’s absolutely on the nose.

However, there is always going to be an element of luck. Maybe you had a random flash of insight that everyone else missed, or maybe you just could not see one crucial piece of information. Maybe you made some unfortunate mistakes or starting going down the wrong path for a long time.

Your ability to solve problems will vary a lot with mood, tiredness, health, random variation, etc. Sometimes the best tactic to solve a problem is to leave it for a few hours — or maybe a few days — then come back to it. Obviously this isn’t applicable to an exam, but that’s just life for you.

I realise I’ve written an essay here but I hope it’s useful. Good luck!

### How to Improve on Mathematical Problem-Solving ?

*From Yassine Lassoued*

Solving problems is a skill that you can improve while solving problems. It is not a matter of quantity... Solving 1000 problems a month does not necessarily make you a good problem solver... However, thinking about how you solved a problem and trying to generalise the approach you used to solve more problems and to improve your problem solving heuristics is the key. There are many tricks and techniques that you need to develop in order to improve your problem solving skills:

- Ability to decompose a problem into sub-problems

- Abstraction

- Visual conceptualisation of mathematical concepts

One of my favourite techniques is scepticism. Sometimes it helps when you are asked to prove a theorem to try to find a counter-example to it. By doing so, and (obviously) failing, you may spot the key property (hypothesis) that is making it impossible for you to find such a counter-example. That is exactly where you should search for an answer...

*From Fadil Habibi D*

Short answer: Solve more problems!

Long answer:

In general, one cannot improve his problem-solving skill without solving problems. The problems should not be too easy that even he doesn't need to spend longer time that he usually spends for solving casual problems.

In principle, one need tool(s) to solve math problems. The tools may come in form of theorems, algorithms, techniques, Algebraic identities, and many more. The more advanced the problem, the more complicated the tool(s) or combinations of tools needed to solve it. If you really wish to be able to solve more advanced problems, learning those tools are usually mandatory.

*From Sudarshan Diwanji*

To improve on math problem solving you need to properly understand the problem and also understand the derivation of the formula which is used in the problem.

For example,

You need to find the area the area of a cirlcle.

- Write down all given things like the lenght of the radius.
- The formula for finding the area of a circle is =pi x r x r
- watch some youtube videos and understand how this formula is derived.
- Lastly, dont solve the problem completely but solve only till the concept of the question is cleared.Dont do calculations until nescessary.

When you are done solving some questions on the same topic , try and solve a HOTS(Higher Order Thinking Skills) question. You will get many of these on the net.