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Chapter 5: Problem 7

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

Short Answer

Expert verified
The quotient of the division \(\frac{2y^2 + 5y + 2}{y+2}\) is \(2y + 1\).

Step by step solution

01

Initial Division

The first term of the polynomial, \(2y^2\), is divided by the first term of the divisor, \(y\), to find the first term of the quotient. \(2y^2 \div y = 2y\). So, the first term of the quotient is \(2y\).
02

Multiply and Subtract

Next, the partial quotient \(2y\) is multiplied by the whole divisor \(y+2\), and subtracted from the original polynomial \(2y^2+5y+2\). The calculation is \(2y^2+5y+2 - 2y \cdot (y+2) = 2y^2+5y+2 - (2y^2 + 4y) = y + 2\). So, the new polynomial after the subtraction is \(y + 2\).
03

Repeat Division

The first term of the new polynomial, \(y\), is now divided by y. This equals 1 which becomes the next term of the quotient. The quotient is now \(2y+1\). This is the final quotient because the degree of the remaining polynomial is less than the degree of the divisor. Therefore, the quotient is the final answer of the division.
04

Check Answer

To check that the division is correct, multiply the divisor (\(y+2\)) by the quotient (\(2y+1\)), and add any remainder. The calculation is \((y+2) \cdot (2y + 1) = 2y^2 + y + 4y + 2 = 2y^2 + 5y + 2\). Since this equals the dividend, the division is correct.

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The mad Dr. Frankenstein has gathered enough bits and pieces (so to speak) for \(2^{-1}+2^{-2}\) of his creature-to-be. Write a fraction that represents the amount of his creature that must still be obtained.

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