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Chapter 6: Problem 16

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

Short Answer

Expert verified
The quotient is \(2x^2 - 7x + 14\) with a remainder of \(-20\).

Step by step solution

01

Write down the coefficients

Identify and write down the coefficients of the polynomial Given: \(2x^3 - 3x^2 + 0x + 8\) Coefficients: \([2, -3, 0, 8]\)
02

Write the divisor's root

For synthetic division, use the root of the divisor \((x+2)\). Set \(x+2 = 0\). Therefore, \(x = -2\).
03

Set up the synthetic division

Create a synthetic division table. Place the root \(-2\) on the left and the coefficients \([2, -3, 0, 8]\) on the top: \(\begin{array}{r|cccc} -2 & 2 & -3 & 0 & 8 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{array}\)
04

Perform the synthetic division

1. Bring down the first coefficient (2) directly below the line. 2. Multiply this value by \(-2\) and write the product under the next coefficient (-3). Add the values (vertex-aligned): \(2 \times -2 = -4\); \(-3 + -4 = -7\). 3. Repeat this process: \(-7 \times -2 = 14\); \(0 + 14 = 14\). 4. Finally: \(14 \times -2 = -28\); \(8 + -28 = -20\). Resulting table: \(\begin{array}{r|cccc} -2 & 2 & -3 & 0 & 8 \ \ \ & 2 & -7 & 14 & -20 \ \end{array}\)
05

Interpret the result

The bottom row (except the final number) gives the coefficients of the quotient polynomial, and the final number is the remainder. Therefore: Quotient polynomial: \(2x^2 - 7x + 14\) Remainder: \(-20\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division
Polynomial division is a method for dividing polynomials, similar to long division used in arithmetic. We use it to simplify expressions or find remainders when a polynomial is divided by another polynomial. In this exercise, we use synthetic division, a streamlined version of polynomial division, typically used when dividing by a linear polynomial of the form \(x + a\) or \(x - a\).
This method is quicker and less error-prone but specific to certain divisors. For instance, to divide \(2x^3 - 3x^2 + 8\) by \(x + 2\), we first rewrite it as synthetic division.
Coefficients
Coefficients are the numerical parts of the terms in a polynomial. For the polynomial \(2x^3 - 3x^2 + 8\), the coefficients are [2, -3, 0, 8].
Here, the term \(0x\) acts as a placeholder to account for the missing \(x\) term, ensuring each degree is represented. In synthetic division, we list these coefficients in a table to simplify the division process.
The setup involves placing these coefficients on the top of the table: \2, -3, 0, 8\, ensuring all degrees are covered.
Remainder
The remainder is the amount left over after completing the division process. In synthetic division, the remainder is the last number in the bottom row of our table. For the given problem, after performing the calculations:
\begin{array}{r|cccc} -2 & 2 & -3 & 0 & 8 \ & 2 & -7 & 14 & -20 \ \end{array},
we find the remainder is \-20\. The quotient polynomial is formed from the other numbers in the bottom row, excluding the remainder. Hence, the final result for \(\left(2 x^{3}-3 x^{2}+8\right) \div(x+2)\) is:
Quotient: \2x^2 - 7x + 14\ and Remainder: \-20\.

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Most popular questions from this chapter

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