Remainder Theorem...4

[nycmathguy]

Section 2.3
Question 50

f(x) = -3x^3 + 8x^2 + 10x - 8 with k = 2 + 2sqrt{2}.

Using synthetic division, I got the following:

-3x^2 + (2 - 6sqrt{2})x - (10 - 8sqrt{2})

Remainder = -60 - 8sqrt{2}.

You say?

 

[MathLover1]

post complete question, something is missing here
you have k, and I do not see where and how to use it

 

[nycmathguy]

post complete question, something is missing here
you have k, and I do not see where and how to use it

Look at question 50. I am getting frustrated with Section 2.3. I may have to go back to the beginning of this section.

 

[MathLover1]

50.

write f(x)=(x-k)q(x)+r
demonstrate that f(k)=r

f(x)=-3x^3+8x^2+10x-8
k=2+sqrt(2)

divisor will be factor (x-k)=(x-(2+sqrt(2)))

.....................(-3x^2+(2-3sqrt(2))x+(8- 4sqrt(2))
x-(2+sqrt(2))|-3x^3+8x^2+10x-8
.....................-3x^3+(6+3sqrt(2))x^2
................................(2-3sqrt(2))x^2 +10x
............................ (2-3sqrt(2))x^2 -(2+sqrt(2))(2-3sqrt(2))x...subtract
................................................. 10x+(2+sqrt(2))(2-3sqrt(2))x....simplify
................................................. (10+(-2 - 4sqrt(2))x....simplify, bring down -8
........................................................... (8- 4sqrt(2))x -8
........................................................... (8- 4sqrt(2))x -(2+sqrt(2))(8- 4sqrt(2))
.................................................................................-8 +(2+sqrt(2))(8- 4sqrt(2))....simplify
.................................................................................-8 +8
.......................................................................................0 ->reminder r=0

-3x^3+8x^2+10x-8=((-3x^2+(2-3sqrt(2))x+(8- 4sqrt(2)))*(x-(2+sqrt(2)))

f(k)=r=>f(k)=0
f(2+sqrt(2))= -3(2+sqrt(2))^3+8(2+sqrt(2))^2+10(2+sqrt(2))-8=0

 

[nycmathguy]

50.

write f(x)=(x-k)q(x)+r
demonstrate that f(k)=r

f(x)=-3x^3+8x^2+10x-8
k=2+sqrt(2)

divisor will be factor (x-k)=(x-(2+sqrt(2)))

.....................(-3x^2+(2-3sqrt(2))x+(8- 4sqrt(2))
x-(2+sqrt(2))|-3x^3+8x^2+10x-8
.....................-3x^3+(6+3sqrt(2))x^2
................................(2-3sqrt(2))x^2 +10x
............................ (2-3sqrt(2))x^2 -(2+sqrt(2))(2-3sqrt(2))x...subtract
................................................. 10x+(2+sqrt(2))(2-3sqrt(2))x....simplify
................................................. (10+(-2 - 4sqrt(2))x....simplify, bring down -8
........................................................... (8- 4sqrt(2))x -8
........................................................... (8- 4sqrt(2))x -(2+sqrt(2))(8- 4sqrt(2))
.................................................................................-8 +(2+sqrt(2))(8- 4sqrt(2))....simplify
.................................................................................-8 +8
.......................................................................................0 ->reminder r=0

-3x^3+8x^2+10x-8=((-3x^2+(2-3sqrt(2))x+(8- 4sqrt(2)))*(x-(2+sqrt(2)))

f(k)=r=>f(k)=0
f(2+sqrt(2))= -3(2+sqrt(2))^3+8(2+sqrt(2))^2+10(2+sqrt(2))-8=0

When it comes to long division or synthetic division, can you please post a picture of the math work done on paper for easy reading?

 

[MathLover1]

............(

|

.......................

......................................

.....................................

...subtract
.............................................................

....simplify
...................................................................

...simplify, bring down -8
.....................................................................

.....subtract
....................................................................

.....................................................................

............................................................................................ -8+8
..................................................................................................

 

[nycmathguy]

............(

|

.......................

......................................

.....................................

...subtract
.............................................................

....simplify
...................................................................

...simplify, bring down -8
.....................................................................

.....subtract
....................................................................

.....................................................................

............................................................................................ -8+8
..................................................................................................

Nicely done. How did you do that? LaTex?

 

[MathLover1]

no using wolfram alpha and copy image

 

[nycmathguy]

no using wolfram alpha and copy image

Wolfram works out math problems in full? I guess you pay for advanced service, right?

 

[MathLover1]

no, I do not pay for advanced service

 

[nycmathguy]

no, I do not pay for advanced service

Ok. Cool. I had no idea that wolfram solves problems in step by step fashion.