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Find the value of (k) if the polynomial p(x)=3x^2-7x-2 leaves remainder (- 1) when divided by (x k).?
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Finding the Value of (k) in a Polynomial Division
To find the value of (k) such that the polynomial p(x) = 3x^2 - 7x - 2 leaves a remainder of -1 when divided by (x - k), we need to follow a systematic approach.
Using the Remainder Theorem
- According to the Remainder Theorem, when a polynomial f(x) is divided by (x - a), the remainder is equal to f(a).
- In this case, when p(x) is divided by (x - k), the remainder is -1. This means that p(k) = -1.
Substitute (k) into the Polynomial
- Substitute the value of (k) into the polynomial p(x) and set it equal to -1: p(k) = 3k^2 - 7k - 2 = -1.
Solving the Equation
- Now, solve the equation 3k^2 - 7k - 2 = -1 to find the value of (k).
- Simplify the equation: 3k^2 - 7k - 2 + 1 = 0
- 3k^2 - 7k - 1 = 0
Using the Quadratic Formula
- Use the quadratic formula to solve for (k): k = [-(-7) ± √((-7)^2 - 4*3*(-1))] / 2*3
- k = [7 ± √(49 + 12)] / 6
- k = [7 ± √61] / 6
Calculating the Value of (k)
- Calculate the two possible values of (k) using the formula: k = (7 + √61) / 6 and k = (7 - √61) / 6.
- Therefore, the values of (k) that satisfy the condition are k = (7 + √61) / 6 and k = (7 - √61) / 6.
To find the value of (k) such that the polynomial p(x) = 3x^2 - 7x - 2 leaves a remainder of -1 when divided by (x - k), we need to follow a systematic approach.
Using the Remainder Theorem
- According to the Remainder Theorem, when a polynomial f(x) is divided by (x - a), the remainder is equal to f(a).
- In this case, when p(x) is divided by (x - k), the remainder is -1. This means that p(k) = -1.
Substitute (k) into the Polynomial
- Substitute the value of (k) into the polynomial p(x) and set it equal to -1: p(k) = 3k^2 - 7k - 2 = -1.
Solving the Equation
- Now, solve the equation 3k^2 - 7k - 2 = -1 to find the value of (k).
- Simplify the equation: 3k^2 - 7k - 2 + 1 = 0
- 3k^2 - 7k - 1 = 0
Using the Quadratic Formula
- Use the quadratic formula to solve for (k): k = [-(-7) ± √((-7)^2 - 4*3*(-1))] / 2*3
- k = [7 ± √(49 + 12)] / 6
- k = [7 ± √61] / 6
Calculating the Value of (k)
- Calculate the two possible values of (k) using the formula: k = (7 + √61) / 6 and k = (7 - √61) / 6.
- Therefore, the values of (k) that satisfy the condition are k = (7 + √61) / 6 and k = (7 - √61) / 6.
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Find the value of (k) if the polynomial p(x)=3x^2-7x-2 leaves remainder (- 1) when divided by (x k).?
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Find the value of (k) if the polynomial p(x)=3x^2-7x-2 leaves remainder (- 1) when divided by (x k).? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Find the value of (k) if the polynomial p(x)=3x^2-7x-2 leaves remainder (- 1) when divided by (x k).? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the value of (k) if the polynomial p(x)=3x^2-7x-2 leaves remainder (- 1) when divided by (x k).?.
Find the value of (k) if the polynomial p(x)=3x^2-7x-2 leaves remainder (- 1) when divided by (x k).? for Class 9 2025 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Find the value of (k) if the polynomial p(x)=3x^2-7x-2 leaves remainder (- 1) when divided by (x k).? covers all topics & solutions for Class 9 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the value of (k) if the polynomial p(x)=3x^2-7x-2 leaves remainder (- 1) when divided by (x k).?.
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