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Chapter 2: Problem 28

Use synthetic division to divide. \(\left(5 x^{3}+18 x^{2}+7 x-6\right) \div(x+3)\)

Short Answer

Expert verified
The result of the synthetic division is \(5x^2-3x-2\) with a remainder of \(-12\). This is represented as \(5x^3+18x^2+7x-6 = (x+3)(5x^2-3x-2) - 12\)

Step by step solution

01

Setting Up the Synthetic Division Table

For synthetic division, write down the coefficients of the divisor \(5x^3, 18x^2, 7x, -6\) in a row. Leave a little space for workings, and a space for each degree below 3. The zero coefficient would represent the degree below 3, if the polynomial is not already in standard form (from highest degree to lowest degree). Then, write down the value that makes the divisor zero on the left hand side below this row. For \(x+3\), this is \(-3\).
02

Perform Synthetic Division

To carry out the synthetic division: 1. Draw a line underneath the coefficients, then drop down the first coefficient (which is \(5\)). 2. Multiply \(-3\) (the value from step 1) by the value just written below the line, and write this in the next column underneath the line. 3. Add the numbers in this column and write the result underneath. Repeat these steps until all columns are filled.
03

Interpreting the Result

The numbers on the bottom row of the table correspond to the coefficients of the quotient polynomial, but each term is one degree less than in the original polynomial. The remainder is the last number. In this context, we started with a degree 3 polynomial, so the first number under the line is the coefficient for \(x^2\), the next number is the coefficient for \(x\), and so forth.

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Most popular questions from this chapter

Use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. \(g(x)=\frac{1+3 x^{2}-x^{3}}{x^{2}}\)

(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. \(x^{2}+b x+4=0\)

Solve the inequality and graph the solution on the real number line. \(x^{2}+x<6\)

Solve the inequality and graph the solution on the real number line. \(-2 x^{2}+6 x+15 \leq 0\)

Find the domain of \(x\) in the expression. Use a graphing utility to verify your result. \(\sqrt{\frac{x}{x^{2}-2 x-35}}\)
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