Division of Polynomials

Introduction

Key Concepts

Understanding Polynomials

A polynomial is an algebraic expression consisting of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, $P(x) = 4x^3 - 3x^2 + 2x - 5$ is a polynomial of degree three.

Types of Polynomial Division

There are two primary methods for dividing polynomials:

• Long Division: Similar to numerical long division, this method involves dividing the dividend by the divisor step by step.
• Synthetic Division: A simplified form of polynomial division, applicable when the divisor is a linear binomial of the form $x - c$.

Long Division of Polynomials

Long division involves dividing the highest degree term of the dividend by the highest degree term of the divisor to find the first term of the quotient. This term is then multiplied by the entire divisor, and the result is subtracted from the dividend. The process repeats with the new polynomial until the degree of the remainder is less than the degree of the divisor.

Example:
Divide $P(x) = 2x^3 + 3x^2 - 5x + 6$ by $D(x) = x - 2$.

Step 1: Divide $2x^3$ by $x$ to get $2x^2$.
Step 2: Multiply $2x^2$ by $x - 2$ to get $2x^3 - 4x^2$.
Step 3: Subtract $(2x^3 - 4x^2)$ from $2x^3 + 3x^2$ to get $7x^2$.
Step 4: Divide $7x^2$ by $x$ to get $7x$.
Step 5: Multiply $7x$ by $x - 2$ to get $7x^2 - 14x$.
Step 6: Subtract $(7x^2 - 14x)$ from $7x^2 - 5x$ to get $9x$.
Step 7: Divide $9x$ by $x$ to get $9$.
Step 8: Multiply $9$ by $x - 2$ to get $9x - 18$.
Step 9: Subtract $(9x - 18)$ from $9x + 6$ to get $24$.
Result: $2x^2 + 7x + 9$ with a remainder of $24$, expressed as $$\frac{2x^3 + 3x^2 - 5x + 6}{x - 2} = 2x^2 + 7x + 9 + \frac{24}{x - 2}.$$

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form $x - c$. It is particularly efficient for higher-degree polynomials and reduces the complexity of calculations.

Example:
Divide $P(x) = 2x^3 + 3x^2 - 5x + 6$ by $D(x) = x - 2$ using synthetic division.

Step 1: Write down the coefficients of the dividend: $2$, $3$, $-5$, $6$.
Step 2: Place the root of the divisor ($c = 2$) to the left.
Step 3: Bring down the leading coefficient ($2$).
Step 4: Multiply $2$ by $2$ to get $4$ and add to the next coefficient: $3 + 4 = 7$.
Step 5: Multiply $7$ by $2$ to get $14$ and add to the next coefficient: $-5 + 14 = 9$.
Step 6: Multiply $9$ by $2$ to get $18$ and add to the last coefficient: $6 + 18 = 24$.
Result: The quotient is $2x^2 + 7x + 9$ with a remainder of $24$, expressed as $$\frac{2x^3 + 3x^2 - 5x + 6}{x - 2} = 2x^2 + 7x + 9 + \frac{24}{x - 2}.$$

Remainder Theorem

The Remainder Theorem states that when a polynomial $P(x)$ is divided by a binomial of the form $x - c$, the remainder is equal to $P(c)$. This theorem provides a quick way to evaluate the remainder without performing the entire division process.

Example:
Find the remainder when $P(x) = 2x^3 + 3x^2 - 5x + 6$ is divided by $x - 2$.
According to the Remainder Theorem: $$P(2) = 2(2)^3 + 3(2)^2 - 5(2) + 6 = 16 + 12 - 10 + 6 = 24.$$

Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem. It states that $x - c$ is a factor of the polynomial $P(x)$ if and only if $P(c) = 0$. This theorem is instrumental in factoring polynomials and finding their roots.

Example:
Determine if $x - 3$ is a factor of $P(x) = x^3 - 6x^2 + 11x - 6$.
Calculate $P(3) = 3^3 - 6(3)^2 + 11(3) - 6 = 27 - 54 + 33 - 6 = 0$. Since $P(3) = 0$, $x - 3$ is a factor of $P(x)$.

Application of Polynomial Division

Polynomial division is crucial in various applications, including simplifying rational expressions, solving polynomial equations, and performing partial fraction decomposition. It also aids in understanding the behavior of polynomial functions, such as identifying asymptotes and intercepts.

Example:
Simplify the rational expression $$\frac{x^3 - 6x^2 + 11x - 6}{x - 3}.$$ Using polynomial division, we find that $$\frac{x^3 - 6x^2 + 11x - 6}{x - 3} = x^2 - 3x + 2.$$

Division by Non-Monic Divisors

While synthetic division is typically used when the divisor is monic (leading coefficient is 1), long division accommodates divisors with any leading coefficient. This flexibility is essential when dealing with a broader range of polynomial division problems.

Example:
Divide $P(x) = 4x^3 + 8x^2 - 2x + 4$ by $D(x) = 2x - 1$ using long division.
The process involves similar steps to standard long division, adjusted for the non-monic divisor.

Handling Higher-Degree Polynomials

As polynomials increase in degree, division becomes more complex. Mastery of polynomial division techniques is essential for simplifying high-degree expressions and solving complex equations. Breaking down higher-degree polynomials into manageable parts through division facilitates easier analysis and problem-solving.

Common Pitfalls and Tips

• Misalignment of Terms: Ensure that all like terms are properly aligned according to their degrees during division.
• Sign Errors: Pay careful attention to positive and negative signs when subtracting terms.
• Consistent Notation: Maintain consistent notation for variables and exponents to avoid confusion.
• Verification: Always verify the result by multiplying the quotient by the divisor and adding the remainder to ensure it equals the original dividend.

Polynomial Division in Solving Equations

Polynomial division is instrumental in solving higher-degree equations by simplifying them into lower-degree equations or factoring them completely. Once a polynomial is factored, setting each factor equal to zero allows for the determination of the equation's roots.

Example:
Solve $2x^3 + 3x^2 - 5x + 6 = 0$.
Using synthetic division with a known root (e.g., $x = 2$), we find that the polynomial factors as $(x - 2)(2x^2 + 7x + 9) = 0$. Solving $2x^2 + 7x + 9 = 0$ using the quadratic formula yields complex roots.

Illustrative Example: Detailed Step-by-Step Division

Problem: Divide $P(x) = x^4 - 3x^3 + 4x - 12$ by $D(x) = x - 2$ using long division.
Solution:

Step 1: Divide $x^4$ by $x$ to get $x^3$.
Step 2: Multiply $x^3$ by $x - 2$ to get $x^4 - 2x^3$.
Step 3: Subtract $(x^4 - 2x^3)$ from $x^4 - 3x^3$ to get $-x^3$.
Step 4: Bring down the next term: $0x^2$.
Step 5: Divide $-x^3$ by $x$ to get $-x^2$.
Step 6: Multiply $-x^2$ by $x - 2$ to get $-x^3 + 2x^2$.
Step 7: Subtract $(-x^3 + 2x^2)$ from $-x^3 + 0x^2$ to get $-2x^2$.
Step 8: Bring down the next term: $4x$.
Step 9: Divide $-2x^2$ by $x$ to get $-2x$.
Step 10: Multiply $-2x$ by $x - 2$ to get $-2x^2 + 4x$.
Step 11: Subtract $(-2x^2 + 4x)$ from $-2x^2 + 4x$ to get $0x$.
Step 12: Bring down the last term: $-12$.
Step 13: Divide $0x$ by $x$ to get $0$.
Step 14: Multiply $0$ by $x - 2$ to get $0$.
Step 15: Subtract $0$ from $-12$ to get $-12$.
Result: $x^3 - x^2 - 2x$ with a remainder of $-12$, expressed as $$\frac{x^4 - 3x^3 + 4x - 12}{x - 2} = x^3 - x^2 - 2x - \frac{12}{x - 2}.$$

Polynomial Division in Graph Analysis

Understanding polynomial division aids in graphing polynomial functions. By dividing a polynomial by its factors, one can determine the x-intercepts and analyze the end behavior of the graph. This insight is crucial for sketching accurate representations of polynomial functions.

Example:
Given $P(x) = (x - 1)(x + 2)(x - 3)$, performing polynomial division verifies the roots at $x = 1$, $x = -2$, and $x = 3$, providing key points for graphing.

Advanced Applications: Partial Fraction Decomposition

Polynomial division is a preliminary step in partial fraction decomposition, a technique used to break down complex rational expressions into simpler fractions. This method is particularly useful in calculus for integrating rational functions.

Example:
Decompose $$\frac{x^3 + 2x^2 + 3x + 4}{(x - 1)(x + 2)}.$$ First, perform polynomial division to express the rational function as a polynomial plus a proper fraction, then apply partial fraction decomposition to the proper fraction.

Comparison Table

Summary and Key Takeaways