FP2: Roots of polynomial equations

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. :colondollar:

Spoiler

OP

Okay so I did actually use the fact that g(i) = 3, never mind. :colondollar:

Reply 2

the bear

20

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

Reply 3

Integer123

14

Original post by darkforest

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. :colondollar:

Spoiler

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.

Reply 4

yusyus

21

Original post by darkforest

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. :colondollar:

Spoiler

for bi you can also just plug in i and equate real/imaginary parts to find p/q

Reply 5

yusyus

21

Original post by darkforest

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. :colondollar:

Spoiler

and for part two sub in i/2 and you should get that as a remainder