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Chapter 3: Problem 24

Divide using synthetic division. $$\left(x^{5}+4 x^{4}-3 x^{2}+2 x+3\right) \div(x-3)$$

Short Answer

Expert verified
The result of the synthetic division is \(x^{4}+7x^{3}+21x^{2}+66x+198+\frac{600}{x-3}\)

Step by step solution

01

Setup the Synthetic Division

In this step, set up the polynomial for synthetic division. Write the coefficients of the polynomial in descending order of the powers of \(x\) (replacing missing powers with a zero). Also, write the number we're dividing by \( (x-3)\), which will be the number \(3\). So, we write the numbers as follows: \(3 | 1,4,0,-3,2,3 \)
02

Perform Synthetic Division

The first coefficient \(1\) is brought down as the first coefficient of the quotient. Multiply the new number in the bottom row by the number we're dividing by (3 in this case), and write this product under the second number in the top row. Sum up these numbers. Repeat this process of multiplying and adding for all coefficients in the top row. We get the following numbers: \(3 | 1, 7, 21, 66, 198, 600\)
03

Write the Quotient Polynomial

Notice that we have lost a degree in our polynomial, it's a degree 4 polynomial now. The proper notation for the resulting polynomial is \(x^{4}+7x^{3}+21x^{2}+66x+198\), and the remaining number \(600\) is our remainder. Therefore the result is the polynomial plus the remainder divided by \(x-3\).

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Most popular questions from this chapter

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2(x-4)^{2}\left(x^{2}-25\right)$$

The heat generated by a stove element varies directly as the square of the voltage and inversely as the resistance. If the voltage remains constant, what needs to be done to triple the amount of heat generated?

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-5 x^{4}+7 x^{2}-x+9$$

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+5 x^{2}-9 x-45$$

What are the zeros of a polynomial function and how are they found?
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