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The polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2?
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The polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 ...
**Finding the Value of a in Each Case**
**Given Polynomials:**
- \(x^3 + 2x^2 - 5ax - 7\)
- \(x^3 + ax^2 - 12x + 6\)
**Divisors:**
- \(x + 1\)
- \(x - 2\)
**Remainders:**
- Let the remainder when the first polynomial is divided by \(x + 1\) be \(R_1\)
- Let the remainder when the second polynomial is divided by \(x - 2\) be \(R_2\)
1. **Finding the Remainder \(R_1\):**
- Substitute \(-1\) for \(x\) in the first polynomial:
\(R_1 = (-1)^3 + 2(-1)^2 - 5a(-1) - 7\)
- Simplify the expression to find \(R_1\)
2. **Finding the Remainder \(R_2\):**
- Substitute \(2\) for \(x\) in the second polynomial:
\(R_2 = 2^3 + a(2)^2 - 12(2) + 6\)
- Simplify the expression to find \(R_2\)
3. **Setting the Remainders Equal:**
- Since the remainders are equal, we have:
\(R_1 = R_2\)
- Substitute the expressions for \(R_1\) and \(R_2\) found in steps 1 and 2
- Solve for \(a\) by equating the two expressions
4. **Calculating the Value of \(a\):**
- By solving the equation obtained in step 3, we can find the value of \(a\)
- This value of \(a\) will satisfy the condition that the remainders are equal
By following these steps and performing the necessary calculations, you can determine the value of \(a\) that satisfies the given conditions for both polynomials.
**Given Polynomials:**
- \(x^3 + 2x^2 - 5ax - 7\)
- \(x^3 + ax^2 - 12x + 6\)
**Divisors:**
- \(x + 1\)
- \(x - 2\)
**Remainders:**
- Let the remainder when the first polynomial is divided by \(x + 1\) be \(R_1\)
- Let the remainder when the second polynomial is divided by \(x - 2\) be \(R_2\)
1. **Finding the Remainder \(R_1\):**
- Substitute \(-1\) for \(x\) in the first polynomial:
\(R_1 = (-1)^3 + 2(-1)^2 - 5a(-1) - 7\)
- Simplify the expression to find \(R_1\)
2. **Finding the Remainder \(R_2\):**
- Substitute \(2\) for \(x\) in the second polynomial:
\(R_2 = 2^3 + a(2)^2 - 12(2) + 6\)
- Simplify the expression to find \(R_2\)
3. **Setting the Remainders Equal:**
- Since the remainders are equal, we have:
\(R_1 = R_2\)
- Substitute the expressions for \(R_1\) and \(R_2\) found in steps 1 and 2
- Solve for \(a\) by equating the two expressions
4. **Calculating the Value of \(a\):**
- By solving the equation obtained in step 3, we can find the value of \(a\)
- This value of \(a\) will satisfy the condition that the remainders are equal
By following these steps and performing the necessary calculations, you can determine the value of \(a\) that satisfies the given conditions for both polynomials.
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The polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2?
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The polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2? 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 polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2?.
The polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2? 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 polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2?.
Solutions for The polynomials 𝑥 3 + 2𝑥 2 − 5𝑎𝑥 − 7 and 𝑥 3 + 𝑎𝑥 2 − 12𝑥 + 6 when divided by x + 1 and x – 2 respectively, leave remainders R1 and R2 respectively. Find the value of a in each of the following cases: i. R1 = R2? in English & in Hindi are available as part of our courses for Class 9.
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