Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The result of the division is \(3x^{2}+\frac{1}{2}x-\frac{1}{4}+\frac{55}{8}*(x-\frac{3}{2})^{-1}\)

Step 1: Set up Synthetic Division

We begin by setting up the synthetic division. Write down the coefficients of the polynomial, (3, -4, 0, 5), on the top of the table. The zero is important because there is no \(x\) term in the original polynomial, and hence its coefficient is 0. Note that the coefficients must be written in descending order of powers of \(x\). Then, write down the number which makes the denominator equal to zero. This number comes from solving \(x-\frac{3}{2}=0\)

Step 2: Perform Synthetic Division

To perform the division, start by bringing down the leading coefficient (3). Multiply \(\frac{3}{2}\) by the number just written under the line (3), and write the result under the next coefficient (-4). Add the numbers in this new column, write the result under the line and repeat the multiplication-addition process until there are no more coefficients. Here is how it looks: \[\begin{array}{r | r r r}\ \frac{3}{2} & 3 & -4 & 0 & 5\ & & \frac{9}{2} & -\frac{1}{4} & \frac{15}{8}\ \hline & 3 & \frac{1}{2} & -\frac{1}{4} & \frac{55}{8} \end{array}\]

Step 3: Interpret the Result

In interpreting the result of synthetic division, it must be borne in mind that we started with a cubic polynomial, and we have divided by a linear polynomial. Hence the result should be one degree lower, a quadratic. The coefficients of this quadratic come from the last row of numbers written under the line. Thus, the quotient is \(3x^{2}+\frac{1}{2}x-\frac{1}{4}\). And the remainder is \(\frac{55}{8}\). Hence, the whole fraction equals to the quotient plus the remainder divided by the divisor.