Using Synthetic and Long Division

Introduction

Key Concepts

1. Overview of Polynomial Division

Polynomial division is a process similar to numerical division, where a polynomial is divided by another polynomial of equal or lower degree. The goal is to express the dividend as the product of the divisor and the quotient, plus a remainder. This method is crucial for simplifying complex rational expressions and solving polynomial equations.

2. Long Division of Polynomials

Long division is a systematic method used to divide one polynomial by another, yielding a quotient and a remainder. It closely mirrors the long division process used with numbers.

1. Arrange both the dividend and the divisor in descending order of degree.
2. Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
3. Multiply the entire divisor by this term and subtract the result from the dividend.
4. Bring down the next term from the dividend and repeat the process until all terms are processed.
5. The final expression is the quotient plus the remainder over the divisor.

Divide $2x^3 - 3x^2 + 4x - 5$ by $x - 2$ using long division.

1. Divide $2x^3$ by $x$ to get $2x^2$.
2. Multiply $2x^2$ by $(x - 2)$ to get $2x^3 - 4x^2$.
3. Subtract to obtain $x^2 + 4x$.
4. Divide $x^2$ by $x$ to get $x$.
5. Multiply $x$ by $(x - 2)$ to get $x^2 - 2x$.
6. Subtract to obtain $6x - 5$.
7. Divide $6x$ by $x$ to get $6$.
8. Multiply $6$ by $(x - 2)$ to get $6x - 12$.
9. Subtract to obtain the remainder $7$.

Thus, $2x^3 - 3x^2 + 4x - 5 = (x - 2)(2x^2 + x + 6) + 7$.

3. Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful when dividing by a linear divisor of the form $(x - c)$. It streamlines the process by reducing the number of steps and minimizing the algebraic manipulation required.

1. Write down the coefficients of the dividend polynomial.
2. Set the divisor $(x - c)$ equal to zero and solve for $c$.
3. Bring down the leading coefficient.
4. Multiply $c$ by the value obtained in the previous step and add it to the next coefficient.
5. Repeat the multiplication and addition until all coefficients are processed.
6. The final numbers represent the coefficients of the quotient and the remainder.

Divide $2x^3 - 3x^2 + 4x - 5$ by $x - 2$ using synthetic division.

1. Write the coefficients: $2$, $-3$, $4$, $-5$.
2. Set $x - 2 = 0$ to find $c = 2$.
3. Bring down the $2$.
4. Multiply $2$ by $2$ to get $4$ and add to $-3$ to get $1$.
5. Multiply $1$ by $2$ to get $2$ and add to $4$ to get $6$.
6. Multiply $6$ by $2$ to get $12$ and add to $-5$ to get $7$.

The quotient is $2x^2 + x + 6$ with a remainder of $7$, confirming the result from long division.

4. Applications of Synthetic and Long Division

Both synthetic and long division are instrumental in various applications, including:

• Factor Theorem: Determining whether a linear binomial is a factor of a polynomial.
• Finding Zeros: Simplifying polynomials to find their roots.
• Simplifying Rational Expressions: Reducing complex fractions to their simplest form.
• Polynomial Graphing: Assisting in plotting and analyzing polynomial functions.

5. Advantages and Limitations

While both division methods are valuable, they each have their strengths and constraints.

• Applicable to any divisor polynomial, not just linear ones.
• Provides a clear step-by-step process.

• Can be time-consuming for high-degree polynomials.
• Requires careful bookkeeping to avoid mistakes.

• Faster and more efficient for linear divisors.
• Simplifies calculations by reducing the number of steps.

• Only applicable for divisors of the form $(x - c)$.
• Less flexible when dealing with higher-degree or non-linear divisors.

6. Theoretical Underpinnings

Both division methods are grounded in the Polynomial Remainder Theorem, which states that when a polynomial $f(x)$ is divided by $(x - c)$, the remainder is $f(c)$. This theorem is pivotal in understanding the relationship between polynomials and their factors.

Moreover, these division techniques facilitate the application of the Factor Theorem, asserting that $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$. This connection is fundamental in solving polynomial equations and analyzing function behavior.

7. Solving Polynomial Equations Using Division

Dividing a polynomial by one of its factors simplifies the equation, making it easier to find all roots. For example, if $(x - 2)$ is a factor of $f(x)$, using synthetic or long division to divide $f(x)$ by $(x - 2)$ yields a quadratic polynomial. This quadratic can then be solved using the quadratic formula or factoring techniques to find the remaining zeros.

Find all zeros of the polynomial $f(x) = 2x^3 - 3x^2 + 4x - 5$ given that $(x - 2)$ is a factor.

1. Divide $f(x)$ by $(x - 2)$ using synthetic division to get the quotient $2x^2 + x + 6$ and remainder $7$.
2. Since the remainder is not zero, $(x - 2)$ is not a factor. However, if the remainder were zero, we would proceed to solve the quadratic equation.
3. Assuming $(x - 2)$ is a factor, set $2x^2 + x + 6 = 0$ and solve using the quadratic formula: $x = \frac{-1 \pm \sqrt{1 - 48}}{4} = \frac{-1 \pm \sqrt{-47}}{4}$.
4. The zeros are $x = 2$, $x = \frac{-1 + \sqrt{47}i}{4}$, and $x = \frac{-1 - \sqrt{47}i}{4}$.

8. Connecting to Rational Functions and Their Zeros

In the context of rational functions, synthetic and long division assist in simplifying expressions to identify horizontal and oblique asymptotes, and to find zeros of both the numerator and the denominator. By factoring and dividing, students can fully understand the behavior of rational functions and solve related equations effectively.

Comparison Table

Summary and Key Takeaways