Synthetic division

Definition

Synthetic division is a streamlined method for dividing a polynomial by a linear divisor of the form $(x - c)$, where $c$ is a constant. It simplifies the process by using only the coefficients of the polynomial, allowing for faster calculations compared to long division. This technique is particularly useful in polynomial algebra as it helps to find roots and factor polynomials efficiently.

5 Must Know Facts For Your Next Test

1. Synthetic division only works when dividing by linear factors of the form $(x - c)$, making it simpler than traditional polynomial long division.
2. The coefficients of the polynomial are used directly in synthetic division, eliminating the need to write out all terms of the polynomial explicitly.
3. The last number obtained after completing synthetic division represents the remainder of the division, while the preceding numbers represent the coefficients of the quotient polynomial.
4. If a polynomial is divisible by $(x - c)$, then the remainder will be zero, indicating that $c$ is a root of the polynomial.
5. Synthetic division is often used in conjunction with the Remainder Theorem to quickly evaluate polynomials at specific values.

Review Questions

• How does synthetic division compare to traditional long division when dividing polynomials?
  • Synthetic division is generally faster and simpler than traditional long division because it only involves the coefficients of the polynomial rather than writing out all terms. In synthetic division, you perform calculations with fewer steps and can directly work with the value of $c$ in the divisor $(x - c)$. This makes it a preferred method when dealing with polynomials, especially when you need to find roots or simplify expressions quickly.
• In what situations would you choose synthetic division over other methods of dividing polynomials?
  • Synthetic division is most suitable when you are dividing a polynomial by a linear factor of the form $(x - c)$. It's particularly advantageous when you want to quickly determine if a given value is a root or when you're factoring polynomials. By using synthetic division, you can efficiently obtain both the quotient and remainder without the extra steps required in long division, which saves time and reduces potential errors in calculations.
• Evaluate how synthetic division can be applied alongside the Remainder Theorem to solve polynomial equations.
  • Synthetic division, combined with the Remainder Theorem, allows for quick evaluations of polynomials at specific values. When you divide a polynomial $f(x)$ by $(x - c)$ using synthetic division, you can instantly see if $c$ is a root by checking if the remainder is zero. This not only confirms whether $c$ satisfies $f(x) = 0$, but also provides an efficient way to factor or simplify $f(x)$ further. Hence, synthetic division becomes a powerful tool for both solving equations and exploring polynomial properties.

Related terms

Polynomial: An expression consisting of variables raised to whole number exponents and their coefficients, 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 $f(c)$.

Factoring: The process of rewriting a polynomial as the product of its factors, which can include linear polynomials and irreducible higher-degree polynomials.