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Chapter 5: Problem 10

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{x^{2}-7 x+5}{x+3}$$

Short Answer

Expert verified
The result of the division is \(x - 4\) with a remainder of \(17\). This was verified by showing that \((x + 3)(x - 4) + 17 = x^2 - x + 5\), which matches the original dividend.

Step by step solution

01

Polynomial division: Step 1 - Dividing the first term of the dividend by first term of divisor

Divide the first term of the dividend, \(x^2\), by the first term of the divisor \(x\). The result is \(x\). This becomes the first term of the quotient.
02

Polynomial division: Step 2 - Multiplying the divisor by the first term of quotient and subtracting result from the dividend

Multiply the divisor, \(x + 3\), by the first term of the quotient, \(x\). The result is \(x^2 + 3x\). Subtract this from the original dividend, \(x^2 - 7x + 5\), to get a new temporary dividend, \(-4x + 5\).
03

Polynomial division: Step 3 - Dividing the first term of the temporary dividend by first term of divisor

Divide the first term of the new temporary dividend, \(-4x\), by the first term of the divisor, \(x\). The result, \(-4\), is added to the quotient, resulting in a final quotient of \(x - 4\).
04

Polynomial division: Step 4 - Multiplying the divisor by the new term of the quotient and subtracting it from the temporary dividend

Multiply the divisor, \(x + 3\), by the new term in the quotient, \(-4\), to get \(-4x - 12\). Subtract this from the temporary dividend, \(-4x + 5\), to get a remainder of \(17\).
05

Validate the result

Check the result by showing that the product of the divisor and the quotient plus the remainder equals the original dividend, i.e., \((x + 3)(x - 4) + 17 = x^2 - 4x + 3x -12 + 17 = x^2 - x + 5\), which is identical to the original dividend.

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