Given that f(x) is a polynomial of degree 3, and its remainders are 2x - 5 and 3x + 4 when divided by
x^2- 1 and x^ - 4 respectively. Find the f(x).
Answers
Given info : f(x) is a three degree polynomial and it remainders are (2x - 5) and (3x + 4) when divided by (x² - 1) and (x² - 4) respectively.
To find : the polynomial f(x).
solution : case 1 : dividend = f(x) , divisor = (x² - 1), remainder = (2x - 5)
as f(x) is three degree polynomial, we can assume quotient = (ax + b) [ linear ]
Now, dividend = divisor × quotient + remainder
f(x) = (x² - 1)(ax + b) + (2x - 5)
= ax³ + bx² - ax - b + 2x - 5
= ax³ + bx² + (-a + 2)x + (-b - 5) ......(1)
Case 2 : dividend = f(x), divisor = (x² - 4) , remainder = (3x + 4)
Let quotient = (cx + d )
So, f(x) = (x² - 4)(cx + d) + (3x + 4)
⇒f(x) = cx³ + dx² - 4cx - 4d + 3x + 4
= cx³ + dx² + (-4c + 3)x + (-4d + 4) ......(2)
On comparing (1) to (2) we get,
a = c,.......(i)
d = b, .......(ii)
(-a + 2) = (-4c + 3), ......(iii)
(-b - 5) = (-4d + 4) .........(iv)
from equations (i) and (iii) we get,
3a = 1 ⇒a = 1/3 = c
From equation (ii) and (iv) we get,
3b = 9 ⇒b = 3 = d
Now, f(x) = (x² - 1)(1/3 x + 3) + (2x - 5)
= x³/3 + 3x² - 1/3 x - 3 + 2x - 5
= x³/3 + 3x² + 5/3 x - 8
Therefore the polynomial is x³/3 + 3x² + 5x - 8
Step-by-step explanation:
the answer is x^3/3+ 3x^2+ 5x-8