Long Division

Oculus8596 · 2024-09-19 02:11:44

Divide (x^5 - a^5) by (x - a).

Bob · 2024-09-19 20:16:38

I'm hoping you can do 'ordinary' long division with numbers. The method is similar. It's hard to show the whole process in a thread so I'll just try to get you started.

x^4 +ax^3
_________________________________________________
|
x - a | x^5 + 0x^4 + 0x^3 +0x^2 + 0x -a^5
x^5 - ax^4
______________ -

0 +ax^4 + 0x^3
+ax^4 -a^2x^3

etc

The method is easiest to use if you take great care to keep the powers of x in neat columns. It looks ok on my laptop screen. Hope it stays that way for your screen.

Bob

Oculus8596 · 2024-09-20 10:58:55

Bob wrote:

I'm hoping you can do 'ordinary' long division with numbers. The method is similar. It's hard to show the whole process in a thread so I'll just try to get you started.
x^4 +ax^3
_________________________________________________
|
x - a | x^5 + 0x^4 + 0x^3 +0x^2 + 0x -a^5
x^5 - ax^4
______________ -

0 +ax^4 + 0x^3
+ax^4 -a^2x^3
etc
The method is easiest to use if you take great care to keep the powers of x in neat columns. It looks ok on my laptop screen. Hope it stays that way for your screen.
Bob

The terms in the radicand with 0 as coefficient are called place holders. Is this right? This is also a bit tedious and thus requires extra work to be completed on paper.

Bob · 2024-09-20 19:36:04

When you do numeric long division, you're more likely to succeed with errors if you keep the thousands, hundreds, tens and units neatly in columns. The zero place holders (yes, that's the term) help with the algebraic version. I did this by hand first on a sheet of plain A4 paper. It took about a third of the page to complete.

Bob

Oculus8596 · 2024-09-20 20:51:22

Bob wrote:

When you do numeric long division, you're more likely to succeed with errors if you keep the thousands, hundreds, tens and units neatly in columns. The zero place holders (yes, that's the term) help with the algebraic version. I did this by hand first on a sheet of plain A4 paper. It took about a third of the page to complete.
Bob

It's algebraic practice more than anything else. I don't recall ever working on a problem like this involving the variable "a" as an integer.

Bob · 2024-09-20 21:39:36

A long time ago, when I was preparing for my A level exams (age 18) I did two things for practice.

Starting with an A4 piece of paper I expanded that as a determinant into 27 terms each with three letters.

Then I reversed that using a different order to make, say, this:

which demonstrates a property of determinants.

Later edit: Actually, now I've tried it, one is the negative of the other.

It's hard work and you have to be carefully accurate not to miss a term.

Starting with a quadratic with integers coefficients but which won't factorise easily I would do the formula 'in my head'. ie. remember each calculation answer and carry it forward to the next stage in my head. It's hard work but these two things had two big advantages. (1) It trains the brain to do hard stuff and (2) it teaches you to be very careful with the algebra.

I think of it as similar to an athlete doing weight training. They may not be using weights in their sport but it develops muscle, flexibilty and stamina. Now I'm reminded I'm going to give it a go again to try and keep away from dementia.

Bob

Oculus8596 · 2024-09-20 23:22:35

Bob wrote:

A long time ago, when I was preparing for my A level exams (age 18) I did two things for practice.
Starting with an A4 piece of paper I expanded that as a determinant into 27 terms each with three letters.
Then I reversed that using a different order to make, say, this:
which demonstrates a property of determinants.
It's hard work and you have to be carefully accurate not to miss a term.
Starting with a quadratic with integers coefficients but which won't factorise easily I would do the formula 'in my head'. ie. remember each calculation answer and carry it forward to the next stage in my head. It's hard work but these two things had two big advantages. (1) It trains the brain to do hard stuff and (2) it teaches you to be very careful with the algebra.
I think of it as similar to an athlete doing weight training. They may not be using weights in their sport but it develops muscle, flexibilty and stamina. Now I'm reminded I'm going to give it a go again to try and keep away from dementia.
Bob

This is why I solve math problems. It helps to keep my brain cells alive. I haven't played woth a 3 by 3 matrix in a few years. I need to watch a few video clips to refresh my memory.

Math Is Fun Forum

#1 2024-09-19 02:11:44

Long Division

#2 2024-09-19 20:16:38

Re: Long Division

#3 2024-09-20 10:58:55

Re: Long Division

#4 2024-09-20 19:36:04

Re: Long Division

#5 2024-09-20 20:51:22

Re: Long Division

#6 2024-09-20 21:39:36

Re: Long Division

#7 2024-09-20 23:22:35

Re: Long Division

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