To divide the polynomial \(2x^3 + 3x^2 - 21x + 20\) by \(x - 2\), we can use polynomial long division.
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Setup the Division: We want to divide \(2x^3 + 3x^2 - 21x + 20\) by \(x - 2\).
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Divide the Leading Terms: Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\): \[ 2x^2 \] This will be the first term of the quotient.
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Multiply and Subtract: Multiply \(2x^2\) by \(x - 2\): \[ 2x^2(x - 2) = 2x^3 - 4x^2 \] Now subtract this from the original polynomial: \[ (2x^3 + 3x^2) - (2x^3 - 4x^2) = 3x^2 + 4x^2 = 7x^2 \] Bring down the next term, which is \(-21x\): \[ 7x^2 - 21x \]
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Repeat the Process: Now divide \(7x^2\) by \(x\): \[ 7x \] Multiply \(7x\) by \(x - 2\): \[ 7x(x - 2) = 7x^2 - 14x \] Subtract: \[ (7x^2 - 21x) - (7x^2 - 14x) = -21x + 14x = -7x \] Bring down the next term (constant 20): \[ -7x + 20 \]
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Divide Again: Now divide \(-7x\) by \(x\): \[ -7 \] Multiply \(-7\) by \(x - 2\): \[ -7(x - 2) = -7x + 14 \] Subtract: \[ (-7x + 20) - (-7x + 14) = 20 - 14 = 6 \]
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Final Result: Now we have determined the quotient and the remainder:
- Quotient \(q(x) = 2x^2 + 7x - 7\)
- Remainder \(r(x) = 6\)
Putting it all together, we can express the result as: \[ 2x^2 + 7x - 7 + \frac{6}{x - 2} \]
So, in the requested form, it's: \[ q(x) + \frac{r(x)}{b(x)} = 2x^2 + 7x - 7 + \frac{6}{x - 2} \]