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Chapter 5: Problem 11

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{5 y+10+y^{2}}{y+2}$$

Short Answer

Expert verified
The division \(\frac{5y + 10 + y^2}{y + 2}\) gives quotient \(y + 3\) and remainder \(4\).

Step by step solution

01

Polynomial Division

First, apply polynomial division. Define dividend as \(y^2 + 5y + 10\) and divisor as \(y + 2\). To do this, look for a term that can be multiplied with \(y + 2\) to get the first term of the dividend, which is \(y^2\), and that term is \(y\). So, \(y\) is the first term of the quotient.
02

Subtraction

Next, multiply divisor \(y + 2\) by first term of quotient \(y\) and subtract the result from the dividend. The obtained result becomes the new dividend. After this step, new dividend is \(3y + 10\). Now, repeat the first step and again look for a term that can be multiplied with \(y + 2\) to get the first term of the new dividend \(3y\). So, this term is \(3\) and add it as next term to the quotient.
03

Final Quotient and Remainder

Repeat the process of multiplication and subtraction. The final dividend after subtraction is \(4\). So, \(4\) is the remainder and the quotient of the division becomes \(y + 3\). Therefore, \(\frac{5y + 10 + y^2}{y + 2} = y + 3 + \frac{4}{y + 2}\). Remember, this needs to be check as per problem statement.
04

Validation

To validate the result, multiply the divisor and quotient, and add the remainder. It should be equal to the original dividend. \( (y + 2)(y + 3) + 4 = y^2 + 5y + 10\) , which matches with the original dividend. So, the division is correct.

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