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Chapter 2: Problem 35

Use synthetic division to divide. $$\left(5 x^{3}-6 x^{2}+8\right) \div(x-4)$$

Short Answer

Expert verified
The result of the synthetic division of \(5 x^{3}-6 x^{2}+8\) by \(x - 4\) is \(5x^{2} - 26x + 104 + \frac{424}{x-4}\).

Step by step solution

01

Set Up the Synthetic Division Table

Firstly, write down the coefficients of the polynomial: \(5\), \(-6\), and \(8\). Leave spaces under these numbers for new numbers to be written in. Also leave a space to the left for \(-4\) (the root associated with \(x - 4)\) and have a line drawn across them.
02

Begin the Synthetic Division

Start by bringing down the first coefficient (5) below the line. Then, multiply this number by the root \(-4\), and write the result under the next coefficient, \(-6\). After that, add the numbers in this new column and write the result below the line.
03

Continue the Synthetic Division

Carry on in the same way, multiplying the new number below the line \(-26\) by the root \(-4\) and writing the result under the next coefficient. Then, add the numbers in the column. This is done until you reach the final coefficient.
04

Write the Result as a Polynomial

The numbers below the line are the coefficients of the resulting polynomial. Starting from the highest power which is one degree less than the original polynomial, the result is written like a polynomial. In addition, any remaining number is the remainder, and can be written as a fraction over the divisor (x - 4).

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Most popular questions from this chapter

The maximum safe load uniformly distributed over a one-foot section of a two- inch-wide wooden beam can be approximated by the model $$\text { Load }=168.5 d^{2}-472.1$$ where \(d\) is the depth of the beam. (a) Evaluate the model for \(d=4, d=6, d=8, d=10\) and \(d=12 .\) Use the results to create a bar graph. (b) Determine the minimum depth of the beam that will safely support a load of 2000 pounds.

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$3 x^{2}+b x+10=0$$

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