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Chapter 2: Problem 41

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

Short Answer

Expert verified
The \( \frac{-3 x^{4}}{x-2} = -3x^{3} - 6x^{2} + 12x -24 + \frac{48}{x-2} \).

Step by step solution

01

Step 1. Identify Coefficients of the Polynomial and the Zero of the Divisor

Firstly, list the coefficients of the polynomial \(-3x^{4}\), which are [-3, 0, 0, 0, 0], considering all the terms of the polynomial of degree 4. The zero of the divisor \(x-2\) is 2 because it's the value which makes the divisor equal to zero (when x = 2).
02

Step 2. Set Up the Synthetic Division

For synthetic division, write 2 (the zero from the divisor) on the left side, then set up for synthetic division by drawing a horizontal line under the coefficients, place a vertical line to the right of the zero of the divisor, and write the coefficients [-3, 0, 0, 0, 0] to the right of the line.
03

Perform the Synthetic Division

To perform synthetic division, do as follows: 1. Drop down the leading coefficient (-3 in this case). 2. Multiply the number just written below the line by the zero of the divisor (2), write the product below the next coefficient and add vertically. 3. Repeat the previous process until all the coefficients have been used. After calculation the coefficients will be -3, -6 , 12, -24, 48.
04

Write Down the Result

The result of synthetic division is interpreted in terms of the degree of the original polynomial subtracted by one. For this case, the polynomial \(-3x^{4}\) has been divided by \(x-2\). The resulting coefficients correspond to the polynomial of degree 3. Write the result as \(-3x^{3} - 6x^{2} + 12x -24 + \frac{48}{x-2} \). Note that the last term is the remainder divided by the divisor.

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Most popular questions from this chapter

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=2 x^{4}+5 x^{3}+4 x^{2}+5 x+2$$

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{3}-x^{2}+4 x-4\) Zero \(2 i\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{4}-4 x^{3}+16 x-16\) (a) Upper: \(x=5\) (b) Lower: \(x=-3\)

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$g(x)=2 x^{3}-3 x^{2}-3$$

Determine whether the statement is true or false. Justify your answer. The solution set of the inequality \(\frac{3}{2} x^{2}+3 x+6 \geq 0\) is the entire set of real numbers.
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