Understanding Synthetic Division
Synthetic division is a method used to divide a polynomial by a linear factor of the form (x – k), where k is a constant. This method provides a quick and efficient way to perform long division of polynomials. It is especially useful in finding roots of polynomial functions and simplifying complex algebraic expressions. The synthetic division process involves a set of steps that enable us to factorize polynomials and determine their remainders. In this article, we will explore the process of completing the synthetic division problem for the polynomial 2x^2 + x + 5 when divided by the linear factor (x – 2).
Synthetic Division Step-By-Step
To complete the synthetic division problem, follow these step-by-step instructions:
- Set up the division: Write the coefficients of the polynomial in descending order of degree, including any missing terms with placeholder zeros. The given polynomial is 2x^2 + x + 5, so the corresponding coefficients are 2, 1, and 5.
- Identify the linear factor: The linear factor is in the form (x – k), where k is a constant. In this case, the linear factor is (x – 2).
- Perform the division: Using the coefficients of the polynomial and the value of k, apply the synthetic division process to divide the polynomial.
- Write the quotient and remainder: The result of the synthetic division yields the quotient and remainder, which provide valuable insights into the polynomial’s factorization and roots.
Completing the Given Synthetic Division Problem
Now, let’s complete the synthetic division problem for the polynomial 2x^2 + x + 5 when divided by the linear factor (x – 2).
First, arrange the coefficients of the given polynomial in descending order:
| 2 | 1 | 5 | |
|---|---|---|---|
| Divisor: (x – 2) |
Now, apply the synthetic division process:
| 2 | 1 | 5 | 2 | |
| (2 * 2) = 4 | ||||
| 1 + 4 = 5 |
The result of the synthetic division is as follows:
Quotient: 2x + 5
Remainder: 15
Therefore, when the polynomial 2x^2 + x + 5 is divided by the linear factor (x – 2), the quotient is 2x + 5 and the remainder is 15.
What the Results Indicate
The quotient obtained from the synthetic division represents the result of dividing the given polynomial by the linear factor. In this case, the quotient is 2x + 5. Additionally, the remainder obtained after the division is essential for determining the factorization of the polynomial and identifying its roots.
FAQs
What is synthetic division, and when is it used?
Synthetic division is a method for dividing polynomials by linear factors of the form (x – k). It is used to simplify polynomial division, find roots of polynomial functions, and factorize polynomials efficiently.
Why is it important to set up the division with placeholder zeros for missing terms?
Placeholder zeros ensure that the coefficients of the polynomial are correctly aligned, enabling a smooth and accurate implementation of the synthetic division process.
What information do the quotient and remainder provide in synthetic division?
The quotient obtained from synthetic division represents the result of dividing the polynomial by the linear factor, while the remainder helps in determining the factorization and roots of the polynomial.
Is synthetic division limited to dividing by linear factors?
Yes, synthetic division is specifically designed for dividing polynomials by linear factors of the form (x – k). For division by non-linear factors, alternative methods such as polynomial long division may be used.
By following the steps outlined in this article, you can easily complete the synthetic division problem for the given polynomial and linear factor. The results obtained from synthetic division yield valuable insights into the factorization and roots of the polynomial, making it a powerful tool in algebraic computations. Remember to utilize placeholder zeros for missing terms and to accurately perform the division to obtain the quotient and remainder. With practice, synthetic division can become a fundamental skill in dealing with polynomial expressions and equations.
