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Chapter 5: Problem 24

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{x^{3}+2 x^{2}-3}{x-2}$$

Short Answer

Expert verified
The quotient obtained after dividing \(x^{3}+2 x^{2}-3\) by \(x-2\) is \(x^{2}+4x+8\) with a remainder of \(13\). Product of the divisor and the quotient plus the remainder equals the original function confirming the solution.

Step by step solution

01

Set up the Long Division

The given function is represented as \(\frac{x^{3}+2 x^{2}-3}{x-2}\) and can be set up as long division as follows: \(x^{3}+2 x^{2}+0x-3\) divided by \(x-2\). It is important to note that if any terms are missing from the polynomial, they should be represented as multiplied by zero. In this case, the \(x\) term is missing from the dividend, and hence represented as \(0x\).
02

Perform Polynomial Division

Divide the first term of the dividend (\(x^3\)) by the first term of the divisor (\(x\)) to get \(x^2\) as the first term of the quotient. Multiply the divisor by this term, subtract it from the original dividend to get the new dividend. Repeat the same process till the degree of remainder is less than the degree of divisor or the remainder becomes zero. The process will yield, \(x^{2}+4x+8\) as quotient and a remainder of \(13\).
03

Verification

To check the solution, multiply the divisor (\(x-2\)) with the quotient (\(x^{2}+4x+8\)) and add the remainder (\(13\)). If the result equals the original function (\(x^{3}+2 x^{2}-3\)), then the solution is correct. In this case, the product plus the remainder does equal to the dividend confirming correctness of the solution.

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