Question

What form must the divisor have to make synthetic division an appropriate method for dividing a polynomial? Provide examples to support your claim.

Short answer

The divisor must be of the form \(x - a\) with \(a\) being a real number for synthetic division to be appropriate when dividing a polynomial. It will not work if any other form is used.

Step-by-step solution

Understand Synthetic Division

Synthetic division is a shorthand way of dividing a polynomial by a linear factor of the form \(x - a\) and only by that form. You should understand that synthetic division will not work if the divisor is not a linear monomial.

Explain Why the Divisor Must Be of The Form \(x - a\)

The divisor in synthetic division should be in the form \(x - a\) with \(a\) being a real number because it is a method that simplifies the division process by dealing only with the coefficients of the polynomial. We use \(a\) in the process to divide the coefficients of the polynomial to find the quotient and remainder. Any other type of divisor will not work with the synthetic division method.

Provide examples

Consider a polynomial \(3x^3 - 4x^2 + 5x - 2\). If we want to divide this by a linear factor such as \(x - 2\), synthetic division can be used. But if we want to divide it by something like \(x^2 - 3\), synthetic division cannot be used because the divisor is not of the form \(x - a\).

Key concepts

Polynomial Division

Polynomial division is similar to long division in arithmetic. It involves dividing a polynomial, which is an expression consisting of variables and coefficients, by another polynomial. This process helps in breaking down complex expressions into simpler parts.
Typically, polynomial division is used to divide one polynomial, known as the dividend, by another called the divisor. The goal is to obtain a quotient and possibly a remainder, much like dividing numbers. There are various methods for polynomial division, such as long division and synthetic division.
Long division can be applied to any polynomial division case. However, synthetic division is a more efficient method used specifically for dividing by linear factors, making it faster and easier to perform when applicable. Understanding the type of divisor is key to choosing the correct method.

Linear Factor

A linear factor of a polynomial is a divisor that is a first-degree polynomial of the form \(x - a\), where \(a\) is a real number. Linear factors are important in synthetic division because they simplify the division process.
When using synthetic division, the divisor must be in the form \(x - a\). This allows us to focus solely on the coefficients of the polynomial, utilizing a structured approach to division.
For example:

• Dividing \(x^3 + 2x^2 - 5x + 6\) by \(x - 3\) is appropriate for synthetic division.
• But dividing by \(x^2 - 1\) is not, as it is not a linear factor.

Remember, the linear factor \(x - a\) is essential for the computational ease that synthetic division offers.

Coefficients

In the context of polynomial division, coefficients are the numerical parts of the terms in a polynomial expression. For example, in \(3x^3 - 4x^2 + 5x - 2\), the coefficients are 3, -4, 5, and -2.
Understanding coefficients is crucial in synthetic division because this method simplifies the process by focusing solely on these numbers, ignoring the variables.
Consider this example of synthetic division with the polynomial \(2x^3 - 6x^2 + 2x - 8\) and divisor \(x - 2\):

• Write down the coefficients: 2, -6, 2, and -8.
• Use the number 2 from \(x - 2\) (the root of the divisor) in the division process.
• Perform operations between coefficients based on synthetic division rules.

This focus on coefficients makes synthetic division a quicker, simpler process compared to other methods.

Polynomial Remainder Theorem

The Polynomial Remainder Theorem is a useful tool in polynomial division. It states that when a polynomial \(f(x)\) is divided by a linear factor \(x - a\), the remainder of the division is \(f(a)\). This means you can evaluate \(f(a)\) to find the remainder quickly and easily.
This theorem supports synthetic division by giving insight into what happens during the process, specifically predicting the remainder without completing all steps of traditional division.
For instance, if you divide \(f(x) = 4x^3 - 3x^2 + x - 5\) by \(x - 1\), according to the Polynomial Remainder Theorem, substituting 1 into the polynomial will yield the remainder. That is, \(f(1)\) will show you directly the remainder.
Understanding this theorem can greatly enhance your ability to tackle polynomial division challenges efficiently. It provides a quick check or validation for the results obtained using synthetic division.