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What is the remainder when 7 + 15x – 13x2 + 5x3 is divided by 4 – 3x + x2?
- a)x – 1
- b)x + 1
- c)1 – x
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
What is the remainder when 7 + 15x – 13x2 + 5x3 is divided by 4 ...
Clearly (x - 1) is remainder.
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What is the remainder when 7 + 15x – 13x2 + 5x3 is divided by 4 ...
Polynomial Division
To find the remainder when dividing a polynomial by another polynomial, we can use long division or synthetic division. In this case, we will use long division to divide \(7 + 15x - 13x^2 + 5x^3\) by \(4 - 3x + x^2\).
Long Division
1. Arrange the polynomials in descending order of powers of x:
\(5x^3 - 13x^2 + 15x + 7\) divided by \(x^2 - 3x + 4\).
2. Divide the first term of the dividend by the first term of the divisor:
\(5x^3 / x^2 = 5x\).
3. Multiply the divisor by the result obtained in step 2 and subtract from the dividend:
\(5x(x^2 - 3x + 4) = 5x^3 - 15x^2 + 20x\).
Subtracting this from the dividend gives \(2x^2 - 5x + 7\).
4. Repeat steps 2 and 3 with the new dividend:
\((2x^2 - 5x + 7) / (x^2 - 3x + 4) = 2\).
5. Multiply the divisor by the result obtained in step 4 and subtract from the new dividend:
\(2(x^2 - 3x + 4) = 2x^2 - 6x + 8\).
Subtracting this from the new dividend gives \(-x - 1\).
Remainder
The remainder is \(-x - 1\), which can also be written as \(-x + (-1)\) or \(-1 - x\). Comparing this with the given options, we see that the correct answer is \(x - 1\).
To find the remainder when dividing a polynomial by another polynomial, we can use long division or synthetic division. In this case, we will use long division to divide \(7 + 15x - 13x^2 + 5x^3\) by \(4 - 3x + x^2\).
Long Division
1. Arrange the polynomials in descending order of powers of x:
\(5x^3 - 13x^2 + 15x + 7\) divided by \(x^2 - 3x + 4\).
2. Divide the first term of the dividend by the first term of the divisor:
\(5x^3 / x^2 = 5x\).
3. Multiply the divisor by the result obtained in step 2 and subtract from the dividend:
\(5x(x^2 - 3x + 4) = 5x^3 - 15x^2 + 20x\).
Subtracting this from the dividend gives \(2x^2 - 5x + 7\).
4. Repeat steps 2 and 3 with the new dividend:
\((2x^2 - 5x + 7) / (x^2 - 3x + 4) = 2\).
5. Multiply the divisor by the result obtained in step 4 and subtract from the new dividend:
\(2(x^2 - 3x + 4) = 2x^2 - 6x + 8\).
Subtracting this from the new dividend gives \(-x - 1\).
Remainder
The remainder is \(-x - 1\), which can also be written as \(-x + (-1)\) or \(-1 - x\). Comparing this with the given options, we see that the correct answer is \(x - 1\).
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