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Chapter 2: Problem 41

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

Short Answer

Expert verified
The result of the division \(\frac{-3 x^{4}}{x-2}\) by synthetic division is \(-6x^{3} - 12x^{2} - 24x - 48\).

Step by step solution

01

Setup the Synthetic Division table

In order to conduct synthetic division, we need to set up a table first. In the case of \(-3x^{4}\), there is a missing \(x^{3}\), \(x^{2}\), and \(x\) terms. We will assume them as \(0\). So, our polynomial becomes \(-3x^{4} + 0x^{3} + 0x^{2} + 0x + 0\). Now, set the coefficients of polynomial \(-3x^{4} + 0x^{3} + 0x^{2} + 0x + 0\) to the left of the vertical line and the root of \(x - 2\) which is \(2\) above the horizontal line in the Synthetic Division table.
02

Perform the Synthetic Division

First, bring down the first coefficient which is \(-3\). Then multiply with the root \(2\) and write the answer under the next coefficient, then sum these two values. Repeat this operation until the last coefficient. The last number below the line is the remainder. The other numbers represent the coefficients of the quotient from highest degree to lowest.
03

Writing the Final Answer

The final answer will be the polynomial that the resulting coefficients represent plus the remainder divided by the divisor if there is a remainder. The degree of the polynomial decreases by one degree for each division. In this exercise, since we started from a polynomial degree 4, we will get a polynomial of degree 3 as the result and there would be no remainder.

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Most popular questions from this chapter

The graph of a rational function can never cross one of its asymptotes.

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