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Chapter 6: Problem 11

Use synthetic division to divide. $$ \left(2 x^{3}-x^{2}-7 x+14\right) \div(x+2) $$

Short Answer

Expert verified
The quotient is \(2x^2 - 5x + 3\) with a remainder of 8.

Step by step solution

01

- Set Up Synthetic Division

First, identify the coefficients of the polynomial. For the polynomial given, the coefficients are: 2 (for \(2x^3\)), -1 (for \(-x^2\)), -7 (for \(-7x\)), and 14 (constant term). Also, identify the zero of the divisor \(x + 2\), which is -2.
02

- Write the Coefficients

Write the coefficients in a row: 2, -1, -7, 14. Place the zero of the divisor (which is -2) to the left side.
03

- Begin Synthetic Division

Bring down the first coefficient (2) directly below the line.
04

- Multiply and Add

Multiply -2 (zero of the divisor) by the number you just brought down (2), and write the result under the next coefficient: -2 * 2 = -4. Add this result to the next coefficient (-1): -1 + (-4) = -5.
05

- Repeat Process

Repeat the process: Multiply -2 by -5 to get 10, and add this to the next coefficient (-7): -7 + 10 = 3. Then, multiply -2 by 3 to get -6, and add this to the next coefficient (14): 14 + (-6) = 8.
06

- Write the Quotient and Remainder

The bottom row now reads 2, -5, 3, 8. These numbers represent the coefficients of the quotient polynomial and the remainder. The quotient polynomial is \(2x^2 - 5x + 3\) and the remainder is 8.

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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 used to divide a polynomial by another polynomial of lower degree. It's similar to long division with numbers, but we work with polynomial expressions instead. In our example, we are dividing \((2x^3 - x^2 - 7x + 14)\) by \( (x + 2)\). There are different methods for polynomial division, including long division and synthetic division.
  • Long Division: It involves writing out divisions and remainders, just like traditional long division.
  • Synthetic Division: This is quicker and simpler for dividing polynomials when the divisor is a linear polynomial (in the form \(x - a\)).
For synthetic division, we only use the coefficients of the polynomials involved. This reduces the complexity and makes it easy to solve.
quotient and remainder
When dividing polynomials, the result consists of a quotient and a remainder. The quotient is the polynomial that you get from the division, and the remainder is what's left over. In our example:
  • The quotient polynomial is found by using synthetic division which gives us \((2x^2 - 5x + 3)\).
  • The remainder is the value that doesn鈥檛 nicely divide by the divisor, which in this case is 8.
Understanding the quotient and remainder is crucial because:
  • It helps us rewrite the original division in the form \( \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \).
  • This form provides a way to verify the result of the division.
Hence, if we multiply \((x+2) \) by the quotient \((2x^2 - 5x + 3) \) and add the remainder 8, we get back the original polynomial \((2x^3 - x^2 - 7x + 14) \).
coefficients in algebra
Coefficients are the numerical parts of the terms in a polynomial. In the polynomial \(2x^3 - x^2 - 7x +14 \):
  • 2 is the coefficient of \(x^3\)
  • -1 is the coefficient of \(-x^2 \)
  • -7 is the coefficient of \(-7x \)
  • 14 is a constant term (also a coefficient).
In synthetic division, we use these coefficients and the zero of the divisor to perform simplified calculations. For example, in the given exercise:
  • The coefficients are [2, -1, -7, 14]
  • The zero of the divisor \((x+2)\) is -2
The synthetic division process involves:
  • Bringing down the first coefficient directly.
  • Multiplying the zero of the divisor by each successive value brought down.
  • Adding the result to the next coefficient in the polynomial.
This process continues until all coefficients have been used. The final row of numbers represents the coefficients of the quotient polynomial, with the last number being the remainder.

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