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Polynomials and Partial Fractions: Given f(X)=g(X).(x-1)(x-2)(x-3)...
- Thread starter prowise
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D
Deleted member 4993
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Please help me to find the steps leading to the answer :
Given f(X)=g(X).(x-1)(x-2)(x-3). If f(x) is divided by (x-1), (x-2) and (x-3), the remainders are 4,-1 and 2 respectively.
Using partial fractions, find an expression for f(x).
Thank you.
Is x ≡ X?
What does "." mean? Multiplication?
Okay, so you're given this problem:
\(\displaystyle f\left(x\right)=g\left(x\right)\left(x-1\right)\left(x-2\right)\left(x-3\right)\)
You're told that if you divide f(x) by (x - 1) you have a remainder of 4. Well, what happens if you do that division? What's left over? What does "remainder 4" mean in this context? Similarly, you're told that if you divide f(x) by (x - 2) you have a remainder of -1. What happens if you do that division? What's left over? What does "remainder -1" mean in this context? Repeat the process once more for dividing by (x - 3). Now that you have all of the information from the problem, what do you think your next step should be?
\(\displaystyle f\left(x\right)=g\left(x\right)\left(x-1\right)\left(x-2\right)\left(x-3\right)\)
You're told that if you divide f(x) by (x - 1) you have a remainder of 4. Well, what happens if you do that division? What's left over? What does "remainder 4" mean in this context? Similarly, you're told that if you divide f(x) by (x - 2) you have a remainder of -1. What happens if you do that division? What's left over? What does "remainder -1" mean in this context? Repeat the process once more for dividing by (x - 3). Now that you have all of the information from the problem, what do you think your next step should be?
D
Deleted member 4993
Guest
Okay, so you're given this problem:
\(\displaystyle f\left(x\right)=g\left(x\right)\left(x-1\right)\left(x-2\right)\left(x-3\right)\)
You're told that if you divide f(x) by (x - 1) you have a remainder of 4. Well, what happens if you do that division? What's left over? What does "remainder 4" mean in this context? Similarly, you're told that if you divide f(x) by (x - 2) you have a remainder of -1. What happens if you do that division? What's left over? What does "remainder -1" mean in this context? Repeat the process once more for dividing by (x - 3). Now that you have all of the information from the problem, what do you think your next step should be?
I must be interpreting it wrong!!
According to the statement \(\displaystyle f\left(x\right)=g\left(x\right)\left(x-1\right)\left(x-2\right)\left(x-3\right)\) one of the factors of f(x) is (x-1)
Then, if we divide:
\(\displaystyle g\left(x\right)\left(x-1\right)\left(x-2\right)\left(x-3\right)\)
by (x-1)
The remainder should be 0. Similarly for (x-2) and (x-3).
I am Khanfused now.....
I think i know how to solve it. Thanks for all your contribution. The final answer is 4x^2 - 17x + 17.
Step are as follows :
g(x)= f(x) / [(x-1)(x-2)(x-3)]
By Partial Fractions :
Then f(x) / [(x-1)(x-2)(x-3)] = A/(x-1) + B/(x-2) + C/(x-3)
f(x) = A(x-2)(x-3) + B(x-1)(x-3) +C(x-1)(x-2)
Let x=1, f(1) = A(1-2)(1-3)
4 = A(-1)(-2) --> A=2
Let x=2, f(2).... --> B=1
Let x=3, f(3).... --> C=1
.
.
.
f(x) = 4x^2 - 17x + 17.
Please check for me if my final answer is correct. Feel free to give comments.
Thank you.
Step are as follows :
g(x)= f(x) / [(x-1)(x-2)(x-3)]
By Partial Fractions :
Then f(x) / [(x-1)(x-2)(x-3)] = A/(x-1) + B/(x-2) + C/(x-3)
f(x) = A(x-2)(x-3) + B(x-1)(x-3) +C(x-1)(x-2)
Let x=1, f(1) = A(1-2)(1-3)
4 = A(-1)(-2) --> A=2
Let x=2, f(2).... --> B=1
Let x=3, f(3).... --> C=1
.
.
.
f(x) = 4x^2 - 17x + 17.
Please check for me if my final answer is correct. Feel free to give comments.
Thank you.