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Chapter 5: Problem 26

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

Short Answer

Expert verified
The quotient is \(y^{2} - y\), and the remainder is 1. We have verified this by multiplying the quotient by the divisor and adding the remainder, and found it equals the original numerator.

Step by step solution

01

Division

Start by dividing the first term of the numerator \(2y^{3}\) by the first term of the denominator \(2y\), which results in \(y^{2}\). This will be the first term in the quotient
02

Multiplication

Next, multiply the entire denominator \(2y + 1\) by the term just obtained \(y^{2}\), which gives \(2y^{3} + y^{2}\). Write this below the original numerator and subtract it from the original numerator
03

Subtraction

Performing subtraction, \(2y^{3} - 2y^{3}\) gives 0 and \(-y^{2} - y^{2}\) gives \(-2y^{2}\). Bring down the next term of the numerator which is \(3y\). The new polynomial becomes \(-2y^{2} + 3y\)
04

Division

Repeat the division step. Divide \(-2y^{2}\) by \(2y\) which gives \(-y\). This is the next term in the quotient.
05

Multiplication and subtraction

Take this new quotient and multiply it by the divisor, then subtract the result from the new polynomial (like in steps 2 and 3). This will give a new remainder.
06

Final Quotient and Remainder

Continue this process until we cannot get any new terms in the quotient. In this case, the denominator cannot divide any more terms in the numerator, hence our final quotient becomes \(y^{2} - y\) and the remainder is 1.
07

Verification

To verify, multiply the divisor by the quotient and add the remainder. If this equals the original numerator, then the division is correct.

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