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Chapter 5: Problem 14

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{x^{3}+3 x^{2}+5 x+3}{x+1}$$

Short Answer

Expert verified
The quotient of the given division operation is \(x^{2}+2x+1\) with a remainder of 0.

Step by step solution

01

Perform Long Division

Apply long division of polynomials. Treat \(x+1\) as the divisor and \(x^{3}+3 x^{2}+5 x+3\) as the dividend.\n Divide the highest degree term in the divisor (x) into the highest degree term in the dividend (\(x^{3}\)) to start with. The result (\(x^{2}\)) is the first term of the quotient.
02

Multiply The Divisor

Multiply the divisor (\(x+1\)) by that first term of the quotient (\(x^{2}\)) and subtract the result from the dividend, which will give you a new polynomial to work with.
03

Continue with long division

Continue with the division as before with this new polynomial. You'll get \(x+2\) as the second term in your quotient. The process continues until the degree of the remaining term (remainder) is less than that of the divisor. After finishing the division the quotient should be \(x^{2}+2x+1\) and the remainder 0.
04

Checking the answer

To check your answer, use the remainder theorem. The product of the divisor and the quotient should equal the original dividend when the remainder is added. Here, our remainder is 0, so: \((x+1)(x^{2}+2x+1) = x^{3}+3x^{2}+5x+3\), which verifies our solution.

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