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If the polynomial x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k,the remainder comes out to be x+a,find k and a
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Division of Polynomials
When dividing one polynomial by another, we use a process called polynomial long division. This process is similar to long division for numbers, but instead of dividing by a single digit, we divide by a polynomial.
Given Polynomial
The given polynomial is x^4 - 6x^3 + 16x^2 - 25x + 10.
Divisor Polynomial
The divisor polynomial is x^2 - 2x + k.
Finding the Remainder
To find the remainder, we perform polynomial long division. Let's go through the steps:
1. Divide the first term of the dividend (x^4) by the first term of the divisor (x^2). The result is x^2.
2. Multiply the divisor (x^2 - 2x + k) by the result from step 1 (x^2). The product is x^4 - 2x^3 + kx^2.
3. Subtract the product obtained in step 2 from the dividend (x^4 - 6x^3 + 16x^2 - 25x + 10). This gives us a new polynomial: -4x^3 + (16 - k)x^2 - 25x + 10.
4. Repeat steps 1-3 with the new polynomial obtained in step 3:
a. Divide the first term of the new polynomial (-4x^3) by the first term of the divisor (x^2). The result is -4x.
b. Multiply the divisor (x^2 - 2x + k) by the result from step 4a (-4x). The product is -4x^3 + 8x^2 - 4kx.
c. Subtract the product obtained in step 4b from the new polynomial obtained in step 3. This gives us a new polynomial: (8 - 4k)x^2 - (25 + 4x) + 10.
5. Repeat steps 1-3 with the new polynomial obtained in step 4:
a. Divide the first term of the new polynomial ((8 - 4k)x^2) by the first term of the divisor (x^2). The result is 8 - 4k.
b. Multiply the divisor (x^2 - 2x + k) by the result from step 5a (8 - 4k). The product is (8 - 4k)x^2 - (16 - 8k)x + (8k - 4k^2).
c. Subtract the product obtained in step 5b from the new polynomial obtained in step 4. This gives us a new polynomial: (-16 + 8k)x + (8k - 4k^2) + 10.
6. Finally, we have a linear polynomial (-16 + 8k)x + (8k - 4k^2) + 10. Since the divisor is a quadratic polynomial, the remainder must be a linear polynomial.
Remainder: x
From the above steps, we have obtained the remainder as x. This means that the linear polynomial (-16 + 8k)x + (8k - 4k
When dividing one polynomial by another, we use a process called polynomial long division. This process is similar to long division for numbers, but instead of dividing by a single digit, we divide by a polynomial.
Given Polynomial
The given polynomial is x^4 - 6x^3 + 16x^2 - 25x + 10.
Divisor Polynomial
The divisor polynomial is x^2 - 2x + k.
Finding the Remainder
To find the remainder, we perform polynomial long division. Let's go through the steps:
1. Divide the first term of the dividend (x^4) by the first term of the divisor (x^2). The result is x^2.
2. Multiply the divisor (x^2 - 2x + k) by the result from step 1 (x^2). The product is x^4 - 2x^3 + kx^2.
3. Subtract the product obtained in step 2 from the dividend (x^4 - 6x^3 + 16x^2 - 25x + 10). This gives us a new polynomial: -4x^3 + (16 - k)x^2 - 25x + 10.
4. Repeat steps 1-3 with the new polynomial obtained in step 3:
a. Divide the first term of the new polynomial (-4x^3) by the first term of the divisor (x^2). The result is -4x.
b. Multiply the divisor (x^2 - 2x + k) by the result from step 4a (-4x). The product is -4x^3 + 8x^2 - 4kx.
c. Subtract the product obtained in step 4b from the new polynomial obtained in step 3. This gives us a new polynomial: (8 - 4k)x^2 - (25 + 4x) + 10.
5. Repeat steps 1-3 with the new polynomial obtained in step 4:
a. Divide the first term of the new polynomial ((8 - 4k)x^2) by the first term of the divisor (x^2). The result is 8 - 4k.
b. Multiply the divisor (x^2 - 2x + k) by the result from step 5a (8 - 4k). The product is (8 - 4k)x^2 - (16 - 8k)x + (8k - 4k^2).
c. Subtract the product obtained in step 5b from the new polynomial obtained in step 4. This gives us a new polynomial: (-16 + 8k)x + (8k - 4k^2) + 10.
6. Finally, we have a linear polynomial (-16 + 8k)x + (8k - 4k^2) + 10. Since the divisor is a quadratic polynomial, the remainder must be a linear polynomial.
Remainder: x
From the above steps, we have obtained the remainder as x. This means that the linear polynomial (-16 + 8k)x + (8k - 4k
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If the polynomial x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k,the remainder comes out to be x+a,find k and a
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If the polynomial x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k,the remainder comes out to be x+a,find k and a 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 If the polynomial x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k,the remainder comes out to be x+a,find k and a covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the polynomial x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k,the remainder comes out to be x+a,find k and a.
If the polynomial x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k,the remainder comes out to be x+a,find k and a 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 If the polynomial x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k,the remainder comes out to be x+a,find k and a covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the polynomial x^4-6x^3+16x^2-25x+10 is divided by another polynomial x^2-2x+k,the remainder comes out to be x+a,find k and a.
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