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The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x – 3).?
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The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) an...
f(x) = x4 - 2x3 + 3x2 - ax + b
f(1)=5 and f(-1)=19 ( from given question )
14 - 2.13 + 3.12 - a.1 + b =5 and (-1)4 - 2(-1)3 + 3(-1)2 - a(-1) + b=19
=> 1-2+3-a+b=5 and 1+2+3+a+b=19
=>2-a+b=5 and 6+a+b=19
=>-a+b=5-2 and a+b=19-6
=>-a+b=3....(1) and a+b=13....(2)
on subtracting (1) from (2), we get:
a+b-(-a+b)=13-3
a+b+a-b=10
2a=10
a=5
placing a=5 in equation 1, we get,
-5+b=3 , b=8
a=5, b=8
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The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) an...
Given Information:
The polynomial f(x) = x^4 – 2x^3 + 3x^2 – ax + b leaves the remainders of 5 and 19 when divided by (x – 1) and (x + 1) respectively.
To Find:
The values of a and b, and the remainder when f(x) is divided by (x – 3).
Solution:
Let's start by finding the remainder when f(x) is divided by (x – 1).
Remainder when f(x) is divided by (x – 1):
When f(x) is divided by (x – 1), the remainder is given as 5. This can be represented by the equation:
f(1) = 5
Substituting x = 1 in the polynomial, we get:
f(1) = (1)^4 – 2(1)^3 + 3(1)^2 – a(1) + b = 1 – 2 + 3 – a + b = 5
Simplifying the equation, we have:
2 – a + b = 5 ...(1)
Remainder when f(x) is divided by (x + 1):
Similarly, when f(x) is divided by (x + 1), the remainder is given as 19. This can be represented by the equation:
f(-1) = 19
Substituting x = -1 in the polynomial, we get:
f(-1) = (-1)^4 – 2(-1)^3 + 3(-1)^2 – a(-1) + b = 1 + 2 + 3 + a + b = 19
Simplifying the equation, we have:
6 + a + b = 19 ...(2)
Solving Equations (1) and (2) to Find a and b:
To find the values of a and b, we can solve equations (1) and (2) simultaneously.
Adding equations (1) and (2), we get:
(2 – a + b) + (6 + a + b) = 5 + 19
8 + 2b = 24
2b = 24 – 8
2b = 16
b = 8
Substituting the value of b in equation (1), we get:
2 – a + 8 = 5
-a + 10 = 5
-a = 5 – 10
-a = -5
a = 5
Therefore, the values of a and b are a = 5 and b = 8.
The Remainder when f(x) is divided by (x – 3):
Now, we need to find the remainder when f(x) is divided by (x – 3). To do this, we can use the Remainder Theorem.
The Remainder Theorem states that if a polynomial f(x) is divided by (x – c), then the remainder is equal to f(c).
Substituting x = 3 in the polynomial f(x), we get:
f(3) = (3)^4 – 2(3)^3 + 3(3)^2 – a(3)
The polynomial f(x) = x^4 – 2x^3 + 3x^2 – ax + b leaves the remainders of 5 and 19 when divided by (x – 1) and (x + 1) respectively.
To Find:
The values of a and b, and the remainder when f(x) is divided by (x – 3).
Solution:
Let's start by finding the remainder when f(x) is divided by (x – 1).
Remainder when f(x) is divided by (x – 1):
When f(x) is divided by (x – 1), the remainder is given as 5. This can be represented by the equation:
f(1) = 5
Substituting x = 1 in the polynomial, we get:
f(1) = (1)^4 – 2(1)^3 + 3(1)^2 – a(1) + b = 1 – 2 + 3 – a + b = 5
Simplifying the equation, we have:
2 – a + b = 5 ...(1)
Remainder when f(x) is divided by (x + 1):
Similarly, when f(x) is divided by (x + 1), the remainder is given as 19. This can be represented by the equation:
f(-1) = 19
Substituting x = -1 in the polynomial, we get:
f(-1) = (-1)^4 – 2(-1)^3 + 3(-1)^2 – a(-1) + b = 1 + 2 + 3 + a + b = 19
Simplifying the equation, we have:
6 + a + b = 19 ...(2)
Solving Equations (1) and (2) to Find a and b:
To find the values of a and b, we can solve equations (1) and (2) simultaneously.
Adding equations (1) and (2), we get:
(2 – a + b) + (6 + a + b) = 5 + 19
8 + 2b = 24
2b = 24 – 8
2b = 16
b = 8
Substituting the value of b in equation (1), we get:
2 – a + 8 = 5
-a + 10 = 5
-a = 5 – 10
-a = -5
a = 5
Therefore, the values of a and b are a = 5 and b = 8.
The Remainder when f(x) is divided by (x – 3):
Now, we need to find the remainder when f(x) is divided by (x – 3). To do this, we can use the Remainder Theorem.
The Remainder Theorem states that if a polynomial f(x) is divided by (x – c), then the remainder is equal to f(c).
Substituting x = 3 in the polynomial f(x), we get:
f(3) = (3)^4 – 2(3)^3 + 3(3)^2 – a(3)
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The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x – 3).?
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The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x – 3).? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x – 3).? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x – 3).?.
The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x – 3).? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x – 3).? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The polynomial f(x)= x4 – 2x3 3x2 – ax b when divided by (x – 1) and (x 1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x – 3).?.
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