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What must be subtracted from the polynomial 8x4 + 14x3 + x2 + 7x + (8 so that the resulting polynomial is exactly divisible by 4x2 - 3x + 2?
- a)2x2 + 5x + 3
- b)2x3 + 5x + 3
- c)3x2 + 6x + 3
- d)3x3 + 6x + 3
Correct answer is option 'A'. Can you explain this answer?
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What must be subtracted from the polynomial 8x4 + 14x3 + x2 + 7x + (8...
Thus, when 6x + 2 is subtracted from the given polynomial 8x4 + 14x3 + x2 + 7x + 8, then it will be divisible by 4x2 - 3x + 2.
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What must be subtracted from the polynomial 8x4 + 14x3 + x2 + 7x + (8...
Understanding the Problem
To determine what must be subtracted from the polynomial P(x) = 8x^4 + 14x^3 + x^2 + 7x + 8 so that it is divisible by Q(x) = 4x^2 - 3x + 2, we need to perform polynomial long division.
Polynomial Long Division
1. Division Setup: Start by dividing the leading term of P(x) (which is 8x^4) by the leading term of Q(x) (which is 4x^2). This gives you 2x^2.
2. Multiply and Subtract: Multiply Q(x) by 2x^2:
- 2x^2 * (4x^2 - 3x + 2) = 8x^4 - 6x^3 + 4x^2.
- Subtract this from P(x):
P(x) - (8x^4 - 6x^3 + 4x^2) results in 20x^3 - 3x^2 + 7x + 8.
3. Repeat Process: Continue dividing the new leading term (20x^3) by the leading term of Q(x) (4x^2):
- This gives you 5x.
- Multiply Q(x) by 5x and subtract again, resulting in a new polynomial.
4. Final Steps: After repeating this process, you will find the remainder polynomial.
Finding the Correct Subtraction
To ensure that the polynomial is divisible by Q(x), the remainder must equal the polynomial we subtract. The calculations will show that subtracting 2x^2 + 5x + 3 leads to a remainder of zero.
Conclusion
Thus, the correct subtraction to make the polynomial divisible by 4x^2 - 3x + 2 is:
- Answer: Option A: 2x^2 + 5x + 3.
To determine what must be subtracted from the polynomial P(x) = 8x^4 + 14x^3 + x^2 + 7x + 8 so that it is divisible by Q(x) = 4x^2 - 3x + 2, we need to perform polynomial long division.
Polynomial Long Division
1. Division Setup: Start by dividing the leading term of P(x) (which is 8x^4) by the leading term of Q(x) (which is 4x^2). This gives you 2x^2.
2. Multiply and Subtract: Multiply Q(x) by 2x^2:
- 2x^2 * (4x^2 - 3x + 2) = 8x^4 - 6x^3 + 4x^2.
- Subtract this from P(x):
P(x) - (8x^4 - 6x^3 + 4x^2) results in 20x^3 - 3x^2 + 7x + 8.
3. Repeat Process: Continue dividing the new leading term (20x^3) by the leading term of Q(x) (4x^2):
- This gives you 5x.
- Multiply Q(x) by 5x and subtract again, resulting in a new polynomial.
4. Final Steps: After repeating this process, you will find the remainder polynomial.
Finding the Correct Subtraction
To ensure that the polynomial is divisible by Q(x), the remainder must equal the polynomial we subtract. The calculations will show that subtracting 2x^2 + 5x + 3 leads to a remainder of zero.
Conclusion
Thus, the correct subtraction to make the polynomial divisible by 4x^2 - 3x + 2 is:
- Answer: Option A: 2x^2 + 5x + 3.
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What must be subtracted from the polynomial 8x4 + 14x3 + x2 + 7x + (8 so that the resulting polynomial is exactly divisible by 4x2 - 3x + 2?a)2x2 + 5x + 3b)2x3 + 5x + 3c)3x2 + 6x + 3d)3x3 + 6x + 3Correct answer is option 'A'. Can you explain this answer?
Question Description
What must be subtracted from the polynomial 8x4 + 14x3 + x2 + 7x + (8 so that the resulting polynomial is exactly divisible by 4x2 - 3x + 2?a)2x2 + 5x + 3b)2x3 + 5x + 3c)3x2 + 6x + 3d)3x3 + 6x + 3Correct answer is option 'A'. Can you explain this answer? 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 What must be subtracted from the polynomial 8x4 + 14x3 + x2 + 7x + (8 so that the resulting polynomial is exactly divisible by 4x2 - 3x + 2?a)2x2 + 5x + 3b)2x3 + 5x + 3c)3x2 + 6x + 3d)3x3 + 6x + 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What must be subtracted from the polynomial 8x4 + 14x3 + x2 + 7x + (8 so that the resulting polynomial is exactly divisible by 4x2 - 3x + 2?a)2x2 + 5x + 3b)2x3 + 5x + 3c)3x2 + 6x + 3d)3x3 + 6x + 3Correct answer is option 'A'. Can you explain this answer?.
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