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Chapter 2: Problem 46

Use synthetic division to divide. \(\frac{3 x^{3}-4 x^{2}+5}{x-\frac{3}{2}}\)

Short Answer

Expert verified
The quotient is \(3x^2 - 4x - 1\) and the remainder is \(-1\)

Step by step solution

01

Setting Up

Firstly, list down the coefficients of the dividend, which are 3, -4, and 5. We don't have any missing terms in the polynomial, hence we don't need to include any zeros. Write the number that makes \(x - \frac{3}{2}\) equal to zero, which is \(\frac{3}{2}\), outside the synthetic division symbol. The setup should look like this: \(\begin{array}{c|ccc}\frac{3}{2} & 3 & -4 & 5 \\end{array}\)
02

Performing Synthetic Division

First, bring down the first coefficient (3) as it is. Multiply \(\frac{3}{2}\) by 3, which gives you \(\frac{9}{2}\), and write it below the second coefficient (-4). Add the numbers in the second column and write the sum, which would be \(\frac{-8}{2}\) or -4, below the bar. Now, multiply \(\frac{3}{2}\) by -4, resulting in -6, and write it below the third coefficient (5). Total the numbers in the third column which results in -1. Here's the synthetic division after these operations:\(\begin{array}{c|ccc}\frac{3}{2} & 3 & -4 & 5 \& \frac{9}{2} & -4 & -6 \\hline & 3 & -4 & -1\end{array}\)
03

Writing the Quotient Polynomial and Remainder

From step 2, the numbers obtained below the bar are the coefficients of the quotient polynomial, starting from one degree lower than the dividend polynomial (which was a cubic polynomial). So, those coefficients correspond to a quadratic polynomial and constant. Our result is therefore a quadratic polynomial plus the remainder over the divisor. Thus,\(\frac{3 x^{3}-4 x^{2}+5}{x-\frac{3}{2}} = 3x^2 - 4x - 1 + \frac{-1}{x-\frac{3}{2}}\)

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