Synthetic Division

Definition

Synthetic division is a streamlined method used to divide a polynomial by a linear factor of the form $(x - c)$. This technique simplifies the process by focusing on the coefficients of the polynomial, making it quicker and less error-prone than traditional long division. It is particularly useful when working with polynomial functions as it allows for efficient calculations and helps to find roots or factors without needing to write out all the terms explicitly.

5 Must Know Facts For Your Next Test

1. Synthetic division can only be used for dividing polynomials by linear factors, not quadratic or higher-degree polynomials.
2. The process begins by writing down the coefficients of the polynomial, allowing for quick computation of both the quotient and the remainder.
3. If the divisor is $(x - c)$, you bring down the leading coefficient and then multiply and add sequentially across the coefficients.
4. The final result of synthetic division includes the coefficients of the quotient polynomial and any remainder that can be expressed as a fraction with the divisor.
5. Synthetic division is often faster than long division and is widely used in solving polynomial equations and analyzing polynomial functions.

Review Questions

• How does synthetic division differ from traditional long division when dividing polynomials, and why is it preferred in certain situations?
  • Synthetic division differs from traditional long division by focusing solely on the coefficients of the polynomial, which streamlines the process and reduces complexity. Unlike long division, which requires writing out all terms, synthetic division eliminates unnecessary steps, making calculations faster. This method is preferred when dividing by linear factors because it requires less writing and can reduce potential errors during computation, particularly beneficial for larger polynomials.
• Demonstrate how synthetic division can be used to verify if $(x - 2)$ is a factor of a given polynomial, such as $f(x) = 2x^3 - 3x^2 + 4x - 8$.
  • To verify if $(x - 2)$ is a factor using synthetic division, first set up synthetic division with 2 (the root) and the coefficients [2, -3, 4, -8]. Bring down the leading coefficient 2. Then multiply 2 by 2 (the root) and add sequentially: (2 * 2 + -3) gives 1; (1 * 2 + 4) gives 6; (6 * 2 + -8) gives 4. The final remainder is 4. Since the remainder isn't zero, $(x - 2)$ is not a factor of $f(x)$.
• Evaluate how synthetic division aids in solving polynomial equations and finding real roots while connecting it to both the Remainder Theorem and Factor Theorem.
  • Synthetic division simplifies solving polynomial equations by quickly providing both quotient and remainder when dividing by linear factors. According to the Remainder Theorem, if a polynomial $f(x)$ is divided by $(x - c)$, the remainder equals $f(c)$. If this remainder is zero, then $(x - c)$ is a factor according to the Factor Theorem. Thus, synthetic division not only allows for efficient factorization but also aids in identifying real roots, facilitating further analysis and problem-solving for polynomial functions.

Related terms

Polynomial: An expression consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication.

Remainder Theorem: A theorem stating that the remainder of the division of a polynomial $f(x)$ by a linear divisor $(x - c)$ is equal to $f(c)$.

Factor Theorem: A theorem stating that a polynomial $f(x)$ has a factor $(x - c)$ if and only if $f(c) = 0$, which relates to finding zeros of the polynomial.