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#1 2024-05-18 05:50:47

mathxyz
Member
From: Brooklyn, NY
Registered: 2024-02-24
Posts: 1,053

Long Division

The last long division for today. This is tedious work.

Problem 103 on 49 goes like this:


Divide x^3 - a^3 by x - a.


I posted this in a FB math group, and several members responded.

Here is one reply:
 
            x^2 + ax + a^2
          _______________
x - a | x^3               - a^3
            x^3 - ax^2
            —————
                   ax^2      - a^3
                   ax^2 - (a^2)x
                   ———————
                   (a^2)x    - a^3
                   (a^2)x    - a^3
                   ———————
                                          0

Look at this line:


ax^2      - a^3
ax^2 - (a^2)x


This is where I get stuck. I think it is easier to use the difference of cubes to expand x^3 - a^2.


You say?

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#2 2024-05-18 06:06:38

Bob
Administrator
Registered: 2010-06-20
Posts: 10,627

Re: Long Division

'Difference of cubes' ?  I only know difference of two squares.

There's a theorem known as the remainder theorem ... that leads to the factor theorem.

If F(x) = Q(x).(x-a)+ R

If we put x = a we get F(a) = R

This can be useful in getting a remainder quickly.

example. An earlier question was "Find the remainder if (4x^3 - 3x^2 + x + 1) ÷ (x + 2)

a in this case is -2 so put that value into the dividend:

4 times (-2)^3 - 3 times (-2)^2 + (-2) + 1 = -32 -12 -2+ 1 = -46 + 1 = -45

If that is no remainder then R = 0 so F(x) = Q(x) . (x-a)  Putting x=a leads to F(a) = 0.



Example.  x^3 - a^3.  Put x = a and we get a^3 - a^3 = 0.  So (x-a) is a factor.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2024-05-18 06:11:16

mathxyz
Member
From: Brooklyn, NY
Registered: 2024-02-24
Posts: 1,053

Re: Long Division

Bob wrote:

'Difference of cubes' ?  I only know difference of two squares.

There's a theorem known as the remainder theorem ... that leads to the factor theorem.

If F(x) = Q(x).(x-a)+ R

If we put x = a we get F(a) = R

This can be useful in getting a remainder quickly.

example. An earlier question was "Find the remainder if (4x^3 - 3x^2 + x + 1) ÷ (x + 2)

a in this case is -2 so put that value into the dividend:

4 times (-2)^3 - 3 times (-2)^2 + (-2) + 1 = -32 -12 -2+ 1 = -46 + 1 = -45

If that is no remainder then R = 0 so F(x) = Q(x) . (x-a)  Putting x=a leads to F(a) = 0.



Example.  x^3 - a^3.  Put x = a and we get a^3 - a^3 = 0.  So (x-a) is a factor.

Bob


Very informative.  Thanks.

Can we expand x^3 - a^3 using the difference of cubes?


Look on page 44 for the difference of cubes. This allows us to cancel out the binomial x - a.

Yes?

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#4 2024-05-18 19:20:24

Bob
Administrator
Registered: 2010-06-20
Posts: 10,627

Re: Long Division

p44 example 10 has (x-1). But the quesion here shows this can be generalised.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#5 2024-05-19 01:48:35

mathxyz
Member
From: Brooklyn, NY
Registered: 2024-02-24
Posts: 1,053

Re: Long Division

Bob wrote:

p44 example 10 has (x-1). But the quesion here shows this can be generalised.

Bob

On page 44 you can see the difference of cubes formula that can be used to expand x^3 - a^3 in our long division problem.

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#6 2024-05-19 04:23:33

mathxyz
Member
From: Brooklyn, NY
Registered: 2024-02-24
Posts: 1,053

Re: Long Division

Bob wrote:

p44 example 10 has (x-1). But the quesion here shows this can be generalised.

Bob


Bob,

Using formula 5 on page 44, I can easily divide x^3 - a^3 by x - a.


Look:

x^3 - a^3 = (x - a)(x^2 + ax + a^2).


So,  [(x - a)(x^2 + ax + a^2)]/(x - a) = (x^2 + ax + a^2).


Much easier than using long division. However, I do understand why Sullivan instructs students to use long division.

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#7 2024-05-19 04:37:16

Bob
Administrator
Registered: 2010-06-20
Posts: 10,627

Re: Long Division

Easier .... like using the quadratic formula rather than completing the square.  But where did the formula come from?  Mathematicians like to know the answer to that question ... understanding rather than rote learning.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#8 2024-05-19 07:03:48

mathxyz
Member
From: Brooklyn, NY
Registered: 2024-02-24
Posts: 1,053

Re: Long Division

Bob wrote:

Easier .... like using the quadratic formula rather than completing the square.  But where did the formula come from?  Mathematicians like to know the answer to that question ... understanding rather than rote learning.

Bob


I totally get it. Sullivan wants students to practice long division on that particular page.

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