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FP2: Roots of polynomial equations watch

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    I need some help with part (a) and part (b).

    For part (a), I knew that the statement was true as it was stated many times in all of my textbooks, but I didn't know exactly how to prove it.

    The only other way I can think of right now is to say something like "let a=2 (any number)", and then find the other quadratic factor when the polynomial is divided by (x-2), and show that when you expand it out it's equal to finding f(a).

    For part (b), I'm just rather stuck altogether. I figured that I must have to find p and q by solving simultaneous equations, but I can't seem to be able to form them with the information given.

    (As I was just writing this out, I noticed that I probably need to use the statement from part (a) in some way, the wording kind of gives it away...)

    Anyway, here is my working out. I apologise for it being rather messy as I did it all whilst drunk on Christmas Eve.
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    Okay so I did actually use the fact that g(i) = 3, never mind.
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    for part a)

    let f(x) = ( x -a )( x - b )(x - c )....( x - n ) + R

    let x = a...

    f(a) = 0*-b*-c*....*-n + R

    f(a) = R
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    (Original post by darkforest)
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Size:  506.2 KB

    I need some help with part (a) and part (b).

    For part (a), I knew that the statement was true as it was stated many times in all of my textbooks, but I didn't know exactly how to prove it.

    The only other way I can think of right now is to say something like "let a=2 (any number)", and then find the other quadratic factor when the polynomial is divided by (x-2), and show that when you expand it out it's equal to finding f(a).

    For part (b), I'm just rather stuck altogether. I figured that I must have to find p and q by solving simultaneous equations, but I can't seem to be able to form them with the information given.

    (As I was just writing this out, I noticed that I probably need to use the statement from part (a) in some way, the wording kind of gives it away...)

    Anyway, here is my working out. I apologise for it being rather messy as I did it all whilst drunk on Christmas Eve.
    Spoiler:
    Show
    For (a), let the remainder when f(x) is divided by a be r. This means that f(x)=(x-a)p(x)+r, where p(x) is some polynomial. From here, you should be able to show that f(a)=r.

    For (b), use the fact that g(i)=3 as you said and then compare real and imaginary parts to find the values of p and q. From here, part (ii) should be able to be done.
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    (Original post by darkforest)
    Name:  IMG_5322.jpg
Views: 25
Size:  506.2 KB

    I need some help with part (a) and part (b).

    For part (a), I knew that the statement was true as it was stated many times in all of my textbooks, but I didn't know exactly how to prove it.

    The only other way I can think of right now is to say something like "let a=2 (any number)", and then find the other quadratic factor when the polynomial is divided by (x-2), and show that when you expand it out it's equal to finding f(a).

    For part (b), I'm just rather stuck altogether. I figured that I must have to find p and q by solving simultaneous equations, but I can't seem to be able to form them with the information given.

    (As I was just writing this out, I noticed that I probably need to use the statement from part (a) in some way, the wording kind of gives it away...)

    Anyway, here is my working out. I apologise for it being rather messy as I did it all whilst drunk on Christmas Eve.
    Spoiler:
    Show
    for bi you can also just plug in i and equate real/imaginary parts to find p/q
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    (Original post by darkforest)
    Name:  IMG_5322.jpg
Views: 25
Size:  506.2 KB

    I need some help with part (a) and part (b).

    For part (a), I knew that the statement was true as it was stated many times in all of my textbooks, but I didn't know exactly how to prove it.

    The only other way I can think of right now is to say something like "let a=2 (any number)", and then find the other quadratic factor when the polynomial is divided by (x-2), and show that when you expand it out it's equal to finding f(a).

    For part (b), I'm just rather stuck altogether. I figured that I must have to find p and q by solving simultaneous equations, but I can't seem to be able to form them with the information given.

    (As I was just writing this out, I noticed that I probably need to use the statement from part (a) in some way, the wording kind of gives it away...)

    Anyway, here is my working out. I apologise for it being rather messy as I did it all whilst drunk on Christmas Eve.
    Spoiler:
    Show
    and for part two sub in i/2 and you should get that as a remainder
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    Hi all, thanks for the help!

    I think I have a viable solution here:

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    I probably should have read the question more carefully!
 
 
 
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