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Find the values of a and b so that the polynomial (x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)?
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Find the values of a and b so that the polynomial (x3-10x2+ax+b) is ex...
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Find the values of a and b so that the polynomial (x3-10x2+ax+b) is ex...
Solution:
Step 1: Find the Remainder When Dividing by (x - 1)
Step 2: Find the Remainder When Dividing by (x - 2)
Step 3: Set Up and Solve Equations
Given: The polynomial (x3 - 10x2 + ax + b) is exactly divisible by (x - 1) and (x - 2).
Step 1: Find the Remainder When Dividing by (x - 1)
When a polynomial is divided by x - a, the remainder is the value of the polynomial when x = a. So, to find the remainder when dividing (x3 - 10x2 + ax + b) by (x - 1), we substitute x = 1:
R(1) = (13 - 10a + b)
Therefore, (x - 1) divides (x3 - 10x2 + ax + b) if and only if R(1) = 0.
Step 2: Find the Remainder When Dividing by (x - 2)
Similarly, to find the remainder when dividing (x3 - 10x2 + ax + b) by (x - 2), we substitute x = 2:
R(2) = (2a + 2b - 28)
Therefore, (x - 2) divides (x3 - 10x2 + ax + b) if and only if R(2) = 0.
Step 3: Set Up and Solve Equations
To satisfy the conditions of the problem, we need to find values of a and b such that:
- R(1) = 0
- R(2) = 0
Substituting the expressions for R(1) and R(2) from steps 1 and 2, we get:
- 13 - 10a + b = 0
- 2a + 2b - 28 = 0
Solving these two equations simultaneously, we get:
- a = 9
- b = -4
Therefore, the values of a and b that satisfy the conditions of the problem are a = 9 and b = -4.
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Find the values of a and b so that the polynomial (x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)?
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Find the values of a and b so that the polynomial (x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Find the values of a and b so that the polynomial (x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the values of a and b so that the polynomial (x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)?.
Find the values of a and b so that the polynomial (x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Find the values of a and b so that the polynomial (x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the values of a and b so that the polynomial (x3-10x2+ax+b) is exactly divisible by (x-1) as well as (x-2)?.
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