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Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x -k leave the same remainder when divided by (x-1)?
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Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x...
Introduction:
To find the value of 'k' such that the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x - k leave the same remainder when divided by (x-1), we will use the Remainder Theorem.
Remainder Theorem:
The Remainder Theorem states that if a polynomial f(x) is divided by (x-a), the remainder is equal to f(a).
Step 1: Divide the polynomials by (x-1):
We will divide the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x - k by (x-1) using long division.
2x² + (k+2)x + (k+3)
____________________________
x - 1 | 2x³ + kx² + 3x - 5
2x³ - 2x²
___________
(k+2)x² + 3x
(k+2)x² - (k+2)x
_________________
4x - 5
4x - 4
__________
-1
The remainder is -1.
x² - 3x + (k-1)
______________________
x - 1 | x³ + x² - 4x - k
x³ - x²
___________
2x² - 4x
2x² - 2x
__________
-2x - k
-2x + 2
__________
-k - 2
The remainder is -k - 2.
Step 2: Equate the remainders:
Since the two polynomials leave the same remainder when divided by (x-1), we equate the remainders obtained.
-1 = -k - 2
Step 3: Solve for 'k':
To solve for 'k', we will isolate the variable on one side of the equation.
-1 + 2 = -k
1 = -k
k = -1
Conclusion:
The value of 'k' that makes the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x - k leave the same remainder when divided by (x-1) is k = -1.
To find the value of 'k' such that the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x - k leave the same remainder when divided by (x-1), we will use the Remainder Theorem.
Remainder Theorem:
The Remainder Theorem states that if a polynomial f(x) is divided by (x-a), the remainder is equal to f(a).
Step 1: Divide the polynomials by (x-1):
We will divide the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x - k by (x-1) using long division.
2x² + (k+2)x + (k+3)
____________________________
x - 1 | 2x³ + kx² + 3x - 5
2x³ - 2x²
___________
(k+2)x² + 3x
(k+2)x² - (k+2)x
_________________
4x - 5
4x - 4
__________
-1
The remainder is -1.
x² - 3x + (k-1)
______________________
x - 1 | x³ + x² - 4x - k
x³ - x²
___________
2x² - 4x
2x² - 2x
__________
-2x - k
-2x + 2
__________
-k - 2
The remainder is -k - 2.
Step 2: Equate the remainders:
Since the two polynomials leave the same remainder when divided by (x-1), we equate the remainders obtained.
-1 = -k - 2
Step 3: Solve for 'k':
To solve for 'k', we will isolate the variable on one side of the equation.
-1 + 2 = -k
1 = -k
k = -1
Conclusion:
The value of 'k' that makes the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x - k leave the same remainder when divided by (x-1) is k = -1.
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Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x -k leave the same remainder when divided by (x-1)?
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Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x -k leave the same remainder when divided by (x-1)? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x -k leave the same remainder when divided by (x-1)? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x -k leave the same remainder when divided by (x-1)?.
Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x -k leave the same remainder when divided by (x-1)? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x -k leave the same remainder when divided by (x-1)? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the value of ‘k’ if the polynomials 2x³ + kx² + 3x - 5 and x³ + x² - 4x -k leave the same remainder when divided by (x-1)?.
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