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Chapter 5: Problem 8

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{2 y^{2}-13 y+21}{y-3}$$

Short Answer

Expert verified
The quotient when \(2y^2-13y+21\) is divided by \(y-3\) is \(2y - 7\) with no remainder.

Step by step solution

01

Long division

Firstly, perform long division on the given polynomials. Start by dividing the first term of the denominator \(y\) into the first term of the numerator \(2y^2\), which results in a quotient of \(2y\). Then, multiply this quotient by the entire denominator \(y-3\) to get \(2y(y-3) = 2y^2-6y\). Subtract this from the original numerator, which gives a new numerator of \(-13y + 6y + 21 = -7y + 21\). Repeat this step with the new numerator, leading to a new quotient term of \(-7\). Thus, the full quotient of \(\frac{2y^2-13y+21}{y-3}\) is \(2y -7\).
02

Find remainder

To find the remainder, we again multiply the latest quotient term \(-7\) by the entire denominator \(y-3\) to get \(-7(y-3) = -7y +21\). Subtract this from the previous balance of \(-7y + 21\) to get \(0\). So, the remainder is \(0\).
03

Verify solution

Now we check the solution by showing that divisor * quotient + remainder = dividend. The divisor is \(y - 3\), the quotient is \(2y - 7\) and the remainder is \(0\), so \(y-3)\*(2y-7) + 0 = 2y^2-13y+21\) which is indeed the original dividend.

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