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Chapter 5: Problem 9

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{x^{2}-3 x+4}{x+2}$$

Short Answer

Expert verified
The quotient of the given division is \(x - 1\) with a remainder of 2.

Step by step solution

01

Polynomial Long Division

Start dividing \(x^{2}-3 x+4\) by \(x+2\). As per the long division method, check how many times does \(x+2\) go into \(x^2\), that is \(x\). Then multiply \(x+2\) by \(x\) and subtract it from \(x^{2}-3 x+4\) which gives \(x - 2\). Now, bring down the 4, and check how many times \(x + 2\) go into \(x-2 + 4\), which is -1. Now, multiply \(x + 2\) by -1 and subtract to check the remainder.
02

Find the Quotient

The division as performed in step 1 gives the quotient \(x - 1\) and a remainder of 2. Thus, we can re-write the given expression as \((x - 1)(x + 2) + 2\).
03

Check the Solution

The solution has to be verified now. For this, multiply the quotient by the divisor, \(x - 1\) and \(x + 2\) respectively, and then to this product, add the remainder. This gives \(x^2 - x + 2x - 2 + 2 = x^{2} - 3x + 4\). Since this matches the dividend, the solution obtained in step 2 is correct.

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