Question

When dividing a polynomial by \(x-c,\) do you prefer to use long division or synthetic division? Does the value of \(c\) make a difference to you in choosing? Give reasons.

Short answer

Prefer synthetic division for simplicity and speed. The value of \(c\) does not matter as long as \(c\) is a constant.

Step-by-step solution

Understand the Problem

The task is to decide between using long division and synthetic division to divide a polynomial by \(x - c\). Consider whether the choice of \(c\) influences this decision.

Review Long Division

Long division of polynomials involves dividing the terms of the polynomial sequentially. It can be used with any divisor.

Review Synthetic Division

Synthetic division simplifies the division process but is only applicable when dividing by \(x - c\). \(c\) must be a constant for this method.

Evaluate Convenience

Synthetic division is generally quicker and less complex than long division. However, it can only be used if the divisor is \(x - c\) where \(c\) is a constant.

Check the Value of c

Note that synthetic division works for any constant \(c\). The actual numerical value of \(c\) does not affect the choice of synthetic over long division, as synthetic division applies for any constant value.

Make a Decision

Prefer synthetic division when dividing a polynomial by \(x - c\) because it is simpler and faster. The specific value of \(c\) does not influence this preference since synthetic division applies as long as \(c\) is a constant.

Key concepts

long division of polynomials

Long division of polynomials is similar to long division with numbers. It involves a sequence of steps where you divide, multiply, subtract, and bring down terms. This method works with any polynomial divisor and is useful for more complex divisions.

Here are the key steps:

• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by this result.
• Subtract this product from the dividend to find the remainder.
• Repeat the process with the new polynomial (remainder) until you reach a degree lower than the divisor.

While it's a methodical process, it can become lengthy and complex with higher-degree polynomials.

Use long division for any polynomial division, especially when the divisor isn’t in the form of \(x - c\).

synthetic division

Synthetic division is a shortcut method that only works for divisors of the form \(x - c\). It's streamlined and less time-consuming compared to long division.

The steps are as follows:

• Write down the coefficients of the polynomial.
• Write the value of \(c\) from \(x - c\).
• Perform synthetic division operations to get the quotient and remainder.

Synthetic division simplifies the calculation and reduces the steps needed, making it quicker for the appropriate type of polynomials.

However, it requires \(c\) to be a constant and cannot be used if the divisor isn't in the form of \(x - c\).

polynomial simplification

When dividing polynomials, the primary goal is to simplify the expression. Both long division and synthetic division help achieve polynomial simplification, but the choice of method can affect the ease and speed.

Simplifying allows you to:

• Identify roots of polynomials.
• Perform further operations like integration or differentiation more easily.
• Solve complex polynomial equations.

Polynomial simplification breaks down the polynomial into simpler components, aiding in better understanding and manipulation.

division by x - c

The divisor \(x - c\) is a linear polynomial where \(c\) is a constant. This form is crucial in determining whether synthetic division can be applied.

Here’s why \(x - c\) is important:

• It simplifies the polynomial division process.
• Enables the use of synthetic division, saving time.
• Useful in finding polynomial roots.

The specific value of \(c\) does not affect the decision between using synthetic or long division since synthetic division works for any constant.

This method doesn't just make division easier but also often simplifies the polynomial considerably.