How Do You Factor Polynomials? A Comprehensive Guide

Factoring polynomials can seem daunting, but understanding the underlying principles makes the process manageable. how.edu.vn provides expert guidance to simplify complex mathematical concepts, ensuring clarity and success. Master polynomial factorization techniques and boost your problem-solving skills.

1. Understanding the Basics of Factoring

Factoring, in its simplest form, is the process of breaking down a number or an algebraic expression into its constituent parts, or factors. This is a fundamental concept in algebra and is essential for simplifying expressions, solving equations, and understanding the behavior of functions. Understanding the basics, from integer factorization to recognizing common patterns in algebraic expressions, is the first step toward mastering more complex factoring techniques.

1.1. Factoring Integers

Factoring integers involves finding the numbers that divide evenly into a given number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 without leaving a remainder. Prime factorization takes this a step further by expressing a number as a product of its prime factors. For instance, the prime factorization of 12 is 2 × 2 × 3, or 2^2 × 3. This is useful in simplifying fractions, finding the greatest common divisor (GCD), and solving various number theory problems.

1.2. Factoring Algebraic Expressions

Factoring algebraic expressions extends the concept of integer factorization to include variables and coefficients. The goal is to rewrite an expression as a product of simpler terms. Common techniques include:

Greatest Common Factor (GCF): Identifying and factoring out the largest term that divides every term in the expression.
Difference of Squares: Recognizing expressions in the form a^2 – b^2, which can be factored into (a + b)(a – b).
Perfect Square Trinomials: Recognizing expressions in the form a^2 + 2ab + b^2 or a^2 – 2ab + b^2, which can be factored into (a + b)^2 or (a – b)^2, respectively.
Factoring by Grouping: Grouping terms in pairs to find common factors and simplify the expression.

These techniques are essential building blocks for factoring more complex polynomials.

1.3. Why is Factoring Important?

Factoring is not just a mathematical exercise; it has numerous practical applications. In algebra, factoring simplifies complex expressions, making them easier to work with. It is crucial in solving quadratic equations, finding the roots of polynomials, and simplifying rational expressions.

Simplifying Expressions: Factoring can reduce complex expressions to simpler forms, making them easier to understand and manipulate.
Solving Equations: Factoring is a key step in solving many types of equations, particularly quadratic equations and higher-degree polynomial equations.
Graphing Functions: Factoring helps in identifying the x-intercepts (roots) of a function, which are essential for sketching its graph.
Real-World Applications: Factoring is used in various fields, including physics, engineering, and computer science, to model and solve real-world problems.

Understanding the importance of factoring provides a solid foundation for mastering more advanced techniques and applying them in various contexts.

2. Factoring Polynomials: A Step-by-Step Guide

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Factoring polynomials involves breaking them down into simpler factors, which can be linear, quadratic, or higher-degree polynomials. This process is essential for solving polynomial equations, simplifying algebraic expressions, and analyzing the behavior of polynomial functions.

2.1. Understanding Polynomials

A polynomial is an expression of the form:

a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0

where:

x is the variable.
a_n, a_{n-1}, ..., a_1, a_0 are the coefficients, which are constants.
n is a non-negative integer, representing the degree of the term.

Examples of polynomials include:

Linear polynomial: 2x + 3
Quadratic polynomial: x^2 - 5x + 6
Cubic polynomial: 3x^3 + 2x^2 - x + 1

Understanding the structure of polynomials is crucial for applying appropriate factoring techniques.

2.2. Basic Factoring Techniques for Polynomials

Several basic techniques can be used to factor polynomials. These include:

Greatest Common Factor (GCF):
- Identify the largest term that divides every term in the polynomial.
- Factor out the GCF from each term.
- Example: Factor 4x^3 + 8x^2 - 12x
  - The GCF is 4x.
  - Factoring out 4x gives 4x(x^2 + 2x - 3).
Difference of Squares:
- Recognize expressions in the form a^2 - b^2.
- Factor into (a + b)(a - b).
- Example: Factor x^2 - 9
  - This is a difference of squares: x^2 - 3^2.
  - Factoring gives (x + 3)(x - 3).
Perfect Square Trinomials:
- Recognize expressions in the form a^2 + 2ab + b^2 or a^2 - 2ab + b^2.
- Factor into (a + b)^2 or (a - b)^2, respectively.
- Example: Factor x^2 + 6x + 9
  - This is a perfect square trinomial: x^2 + 2(3)x + 3^2.
  - Factoring gives (x + 3)^2.
Factoring by Grouping:
- Group terms in pairs.
- Find common factors within each pair.
- Factor out the common factors.
- Example: Factor x^3 + 2x^2 - 3x - 6
  - Group terms: (x^3 + 2x^2) + (-3x - 6).
  - Factor out common factors: x^2(x + 2) - 3(x + 2).
  - Factor out (x + 2): (x + 2)(x^2 - 3).

2.3. Factoring Quadratic Polynomials

Quadratic polynomials are of the form ax^2 + bx + c, where a, b, and c are constants. Factoring quadratic polynomials involves finding two binomials that multiply to give the original quadratic.

Simple Quadratics (a = 1):
- Look for two numbers that multiply to c and add to b.
- Example: Factor x^2 + 5x + 6
  - Find two numbers that multiply to 6 and add to 5. These are 2 and 3.
  - Factoring gives (x + 2)(x + 3).
Complex Quadratics (a ≠ 1):
- Use the “ac method”:
  - Multiply a and c.
  - Find two numbers that multiply to ac and add to b.
  - Rewrite the middle term using these two numbers.
  - Factor by grouping.
- Example: Factor 2x^2 + 7x + 3
  - ac = 2 × 3 = 6.
  - Find two numbers that multiply to 6 and add to 7. These are 1 and 6.
  - Rewrite the middle term: 2x^2 + x + 6x + 3.
  - Factor by grouping: x(2x + 1) + 3(2x + 1).
  - Factor out (2x + 1): (2x + 1)(x + 3).
Using the Quadratic Formula:
- If the quadratic cannot be easily factored, use the quadratic formula to find the roots:
  - x = frac{-b pm sqrt{b^2 - 4ac}}{2a}
- Example: Solve x^2 + 4x + 1 = 0
  - x = frac{-4 pm sqrt{4^2 - 4(1)(1)}}{2(1)}
  - x = frac{-4 pm sqrt{12}}{2}
  - x = -2 pm sqrt{3}
- The factors are (x - (-2 + sqrt{3}))(x - (-2 - sqrt{3})).

These techniques provide a systematic approach to factoring quadratic polynomials, whether they are simple or complex.

2.4. Factoring Higher-Degree Polynomials

Factoring polynomials of degree three or higher can be more challenging, but several strategies can be employed:

Factoring by Grouping:
- Similar to quadratic polynomials, grouping terms and factoring out common factors can simplify the expression.
- Example: Factor x^3 - 2x^2 - 9x + 18
  - Group terms: (x^3 - 2x^2) + (-9x + 18).
  - Factor out common factors: x^2(x - 2) - 9(x - 2).
  - Factor out (x - 2): (x - 2)(x^2 - 9).
  - Factor the difference of squares: (x - 2)(x + 3)(x - 3).
Rational Root Theorem:
- This theorem helps identify potential rational roots of the polynomial.
- If a polynomial has integer coefficients, every rational root has the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
- Example: Factor 2x^3 - 5x^2 + 4x - 1
  - Factors of the constant term (-1): ±1.
  - Factors of the leading coefficient (2): ±1, ±2.
  - Possible rational roots: ±1, ±1/2.
  - Test these roots using synthetic division or direct substitution to find one that results in a remainder of zero.
  - If x = 1 is a root, then (x - 1) is a factor.
  - Use synthetic division to divide the polynomial by (x - 1) and obtain the remaining quadratic factor.
Synthetic Division:
- A method for dividing a polynomial by a linear factor (x - k).
- Helps to find roots and reduce the degree of the polynomial.
- Example: Divide x^3 - 4x^2 + x + 6 by (x - 2)
  - Set up synthetic division:
```
2 | 1  -4   1   6
|      2  -4  -6
----------------
1  -2  -3   0
```
  - The quotient is x^2 - 2x - 3.
  - The remainder is 0, so (x - 2) is a factor.
  - Factor the quotient: x^2 - 2x - 3 = (x - 3)(x + 1).
  - The original polynomial factors to (x - 2)(x - 3)(x + 1).
Numerical Methods:
- For polynomials that cannot be factored easily, numerical methods such as the Newton-Raphson method can be used to approximate the roots.
- These methods are particularly useful for finding irrational or complex roots.

2.5. Common Mistakes to Avoid

When factoring polynomials, it is essential to avoid common mistakes that can lead to incorrect results. Here are some pitfalls to watch out for:

Forgetting to Factor out the GCF:
- Always look for the greatest common factor first.
- Example: In 4x^2 + 8x, forgetting to factor out 4x would lead to an incomplete factorization.
Incorrectly Applying the Difference of Squares:
- Ensure that the expression is in the form a^2 - b^2 before applying the formula.
- Example: x^2 + 4 is not a difference of squares and cannot be factored using this method.
Misidentifying Perfect Square Trinomials:
- Verify that the expression fits the form a^2 + 2ab + b^2 or a^2 - 2ab + b^2.
- Example: x^2 + 4x + 3 is not a perfect square trinomial.
Making Sign Errors:
- Pay close attention to the signs when factoring, especially when using the quadratic formula or factoring by grouping.
- Example: Incorrectly factoring x^2 - 5x + 6 as (x - 2)(x + 3) instead of (x - 2)(x - 3).
Stopping Too Early:
- Ensure that all factors are fully factored.
- Example: Factoring x^4 - 16 as (x^2 + 4)(x^2 - 4) but not further factoring (x^2 - 4) into (x + 2)(x - 2).

By avoiding these common mistakes, you can improve your accuracy and efficiency in factoring polynomials.

3. Advanced Factoring Techniques

While basic factoring techniques cover many common polynomials, some expressions require more advanced methods. These techniques are particularly useful for higher-degree polynomials, complex expressions, and problems that require creative problem-solving.

3.1. Factoring by Substitution

Factoring by substitution involves replacing a complex expression with a simpler variable to make the factoring process more manageable. This technique is useful when dealing with polynomials that contain repeated expressions or composite functions.

Identify the Repeating Expression:
- Look for a part of the polynomial that appears multiple times.
- Example: Factor (x^2 + 1)^2 + 5(x^2 + 1) + 6
  - The expression (x^2 + 1) is repeated.
Substitute a Variable:
- Let y = x^2 + 1.
- The polynomial becomes y^2 + 5y + 6.
Factor the Simplified Polynomial:
- Factor y^2 + 5y + 6 into (y + 2)(y + 3).
Substitute Back:
- Replace y with (x^2 + 1): (x^2 + 1 + 2)(x^2 + 1 + 3).
- Simplify: (x^2 + 3)(x^2 + 4).

This technique simplifies the factoring process by breaking down complex expressions into more manageable parts.

3.2. Factoring Sum and Difference of Cubes

The sum and difference of cubes are special factoring patterns that apply to expressions in the form a^3 + b^3 and a^3 - b^3. These patterns provide a direct way to factor such expressions into simpler terms.

Sum of Cubes:
- The formula for the sum of cubes is:
  - a^3 + b^3 = (a + b)(a^2 - ab + b^2)
- Example: Factor x^3 + 8
  - Recognize that x^3 + 8 = x^3 + 2^3.
  - Apply the formula: (x + 2)(x^2 - 2x + 4).
Difference of Cubes:
- The formula for the difference of cubes is:
  - a^3 - b^3 = (a - b)(a^2 + ab + b^2)
- Example: Factor 27x^3 - 1
  - Recognize that 27x^3 - 1 = (3x)^3 - 1^3.
  - Apply the formula: (3x - 1)(9x^2 + 3x + 1).

Understanding and applying these formulas can significantly simplify the process of factoring expressions involving cubes.

3.3. Using the Remainder and Factor Theorems

The Remainder and Factor Theorems are powerful tools for finding factors and roots of polynomials. These theorems provide a systematic way to test potential roots and simplify the factoring process.

Remainder Theorem:
- If a polynomial f(x) is divided by (x - k), the remainder is f(k).
- This theorem helps to evaluate the polynomial at a specific value and determine the remainder without performing the division.
- Example: Find the remainder when f(x) = x^3 - 4x^2 + 5x - 2 is divided by (x - 1).
  - Evaluate f(1) = (1)^3 - 4(1)^2 + 5(1) - 2 = 1 - 4 + 5 - 2 = 0.
  - The remainder is 0.
Factor Theorem:
- If f(k) = 0, then (x - k) is a factor of f(x).
- This theorem helps to identify factors of the polynomial by finding values of x that make the polynomial equal to zero.
- Example: Determine if (x - 1) is a factor of f(x) = x^3 - 4x^2 + 5x - 2.
  - Since the remainder when f(x) is divided by (x - 1) is 0, (x - 1) is a factor.
  - Use synthetic division to divide f(x) by (x - 1):
```
1 | 1  -4   5  -2
|      1  -3   2
----------------
1  -3   2   0
```
  - The quotient is x^2 - 3x + 2.
  - Factor the quotient: x^2 - 3x + 2 = (x - 1)(x - 2).
  - The original polynomial factors to (x - 1)(x - 1)(x - 2) = (x - 1)^2 (x - 2).

By using the Remainder and Factor Theorems, you can systematically find factors and roots of polynomials, making the factoring process more efficient.

3.4. Factoring Polynomials with Complex Coefficients

Polynomials with complex coefficients can be factored using similar techniques as those with real coefficients, but with additional considerations for complex numbers. Complex numbers have the form a + bi, where a and b are real numbers and i is the imaginary unit (i^2 = -1).

Finding Complex Roots:
- Use the quadratic formula to find complex roots of quadratic polynomials.
- For higher-degree polynomials, numerical methods or specialized software may be needed to find complex roots.
- Example: Solve x^2 + 2x + 5 = 0
  - Use the quadratic formula:
    - x = frac{-2 pm sqrt{2^2 - 4(1)(5)}}{2(1)}
    - x = frac{-2 pm sqrt{-16}}{2}
    - x = frac{-2 pm 4i}{2}
    - x = -1 pm 2i
  - The roots are -1 + 2i and -1 - 2i.
Factoring with Complex Roots:
- If a + bi is a root, then (x - (a + bi)) is a factor.
- The conjugate a - bi is also a root, and (x - (a - bi)) is a factor.
- Example: Factor x^2 + 2x + 5 using the roots -1 + 2i and -1 - 2i.
  - The factors are (x - (-1 + 2i)) and (x - (-1 - 2i)).
  - Simplify: (x + 1 - 2i)(x + 1 + 2i).

3.5. Utilizing Computer Algebra Systems (CAS)

When dealing with complex or high-degree polynomials, Computer Algebra Systems (CAS) can be invaluable tools. These systems can perform symbolic manipulations, find roots, and factor polynomials that would be difficult or impossible to handle manually.

Popular CAS Software:
- Mathematica: A powerful tool for symbolic computation, numerical analysis, and visualization.
- Maple: Another comprehensive CAS with a wide range of mathematical functions and tools.
- SageMath: An open-source CAS based on Python, suitable for various mathematical tasks.
Using CAS for Factoring:
- Enter the polynomial into the CAS.
- Use the appropriate command to factor the polynomial.
- Example (Mathematica):
  - Factor[x^4 + 3x^3 - 5x^2 - 9x + 10]
  - Output: (x - 1) (x - 2) (x + 1) (x + 5)
- Example (Maple):
  - factor(x^4 + 3*x^3 - 5*x^2 - 9*x + 10);
  - Output: (x - 1) (x - 2) (x + 1) (x + 5)

These advanced techniques and tools can help you tackle even the most challenging factoring problems, providing you with a comprehensive understanding of polynomial factorization.

4. Real-World Applications of Factoring Polynomials

Factoring polynomials is not just an abstract mathematical concept; it has numerous practical applications in various fields, including engineering, physics, computer science, and economics. Understanding these applications can provide a deeper appreciation for the importance of factoring.

4.1. Engineering

In engineering, factoring polynomials is used in designing structures, analyzing circuits, and modeling dynamic systems.

Structural Engineering:
- Factoring is used to analyze the stability and stress distribution in structures such as bridges, buildings, and aircraft.
- Polynomial equations describe the relationships between forces, stresses, and deflections.
- Example: Determining the maximum load a beam can withstand involves solving polynomial equations that describe the beam’s bending behavior.
Electrical Engineering:
- Factoring is used in circuit analysis to determine the behavior of electrical circuits.
- Polynomial equations describe the relationships between voltage, current, and impedance.
- Example: Analyzing the stability of a feedback control system involves finding the roots of the characteristic polynomial, which can be done through factoring.
Control Systems:
- Factoring is used in designing control systems for robots, vehicles, and industrial processes.
- Polynomial equations describe the dynamics of the system and the controller.
- Example: Designing a PID controller involves finding the roots of the characteristic polynomial to ensure the system is stable and responds quickly to changes.

4.2. Physics

In physics, factoring polynomials is used in solving equations related to motion, energy, and quantum mechanics.

Classical Mechanics:
- Factoring is used to solve equations of motion for objects under various forces.
- Polynomial equations describe the relationships between position, velocity, and acceleration.
- Example: Determining the trajectory of a projectile involves solving polynomial equations that describe its motion under gravity.
Quantum Mechanics:
- Factoring is used in solving the Schrödinger equation, which describes the behavior of quantum systems.
- Polynomial equations describe the energy levels and wave functions of particles.
- Example: Finding the energy levels of an electron in a potential well involves solving polynomial equations that arise from the Schrödinger equation.
Thermodynamics:
- Factoring can be applied in thermodynamic calculations, particularly when dealing with equations of state that relate pressure, volume, and temperature.
- Polynomial approximations are often used to model the behavior of real gases and fluids.

4.3. Computer Science

In computer science, factoring polynomials is used in cryptography, coding theory, and algorithm design.

Cryptography:
- Factoring large numbers into their prime factors is a fundamental problem in cryptography.
- Many encryption algorithms, such as RSA, rely on the difficulty of factoring large numbers to ensure security.
- Example: The RSA algorithm involves choosing two large prime numbers and multiplying them together to create a public key. Factoring this public key would allow an attacker to break the encryption.
Coding Theory:
- Factoring is used in designing error-correcting codes, which are used to transmit data reliably over noisy channels.
- Polynomials are used to represent codewords, and factoring helps in designing codes that can detect and correct errors.
- Example: Reed-Solomon codes, which are used in CD players and QR codes, rely on polynomial arithmetic and factoring to correct errors in the data.
Algorithm Design:
- Factoring is used in designing efficient algorithms for various computational problems.
- Polynomial equations describe the relationships between inputs and outputs, and factoring helps in simplifying these relationships.
- Example: Designing an algorithm for polynomial multiplication involves using factoring techniques to reduce the number of operations required.

4.4. Economics

In economics, factoring polynomials is used in modeling supply and demand, analyzing market equilibrium, and forecasting economic trends.

Supply and Demand:
- Polynomial equations are used to model the relationships between supply, demand, and price.
- Factoring helps in finding the equilibrium points where supply equals demand.
- Example: Determining the market equilibrium price and quantity involves solving polynomial equations that describe the supply and demand curves.
Market Equilibrium:
- Factoring is used to analyze the stability of market equilibrium points.
- Polynomial equations describe the dynamics of the market, and factoring helps in determining whether the equilibrium is stable or unstable.
- Example: Analyzing the stability of a competitive market involves finding the roots of the characteristic polynomial, which can be done through factoring.
Economic Forecasting:
- Factoring is used in forecasting economic trends by analyzing historical data and building polynomial models.
- Polynomial equations describe the relationships between economic variables, and factoring helps in identifying patterns and making predictions.
- Example: Forecasting GDP growth involves building polynomial models that describe the relationships between GDP, investment, consumption, and government spending.

These real-world applications demonstrate the broad utility of factoring polynomials in solving complex problems and making informed decisions in various fields.

5. Tips and Tricks for Mastering Factoring

Mastering factoring requires practice, patience, and a strategic approach. Here are some tips and tricks to help you improve your skills and tackle factoring problems with confidence:

5.1. Practice Regularly

Consistent practice is essential for mastering any mathematical skill, and factoring is no exception.

Solve a Variety of Problems:
- Work through a wide range of factoring problems, from basic to advanced, to build your skills and confidence.
- Use textbooks, online resources, and practice worksheets to find problems to solve.
Review Your Mistakes:
- Pay attention to your mistakes and learn from them.
- Identify the types of problems you struggle with and focus on improving those areas.
Set Goals:
- Set achievable goals for your practice sessions, such as solving a certain number of problems or mastering a specific technique.
- Track your progress and celebrate your successes.

5.2. Develop a Strategic Approach

Having a systematic approach to factoring can help you solve problems more efficiently and accurately.

Look for the GCF First:
- Always start by looking for the greatest common factor (GCF).
- Factoring out the GCF simplifies the expression and makes it easier to factor further.
Identify Patterns:
- Recognize common factoring patterns, such as the difference of squares, perfect square trinomials, and sum/difference of cubes.
- Identifying these patterns can help you quickly factor the expression.
Break Down Complex Problems:
- Break down complex factoring problems into smaller, more manageable steps.
- Use techniques such as factoring by grouping, substitution, and the Remainder/Factor Theorems to simplify the expression.
Check Your Work:

After factoring, multiply the factors back together to ensure you get the original polynomial. This is a quick way to catch errors.

5.3. Use Visual Aids

Visual aids can help you understand and remember factoring techniques.

Create Flashcards:
- Create flashcards for common factoring patterns and formulas.
- Use the flashcards to review the patterns and formulas regularly.
Draw Diagrams:
- Draw diagrams to help you visualize the factoring process.
- For example, use a rectangle to represent a quadratic expression and divide it into smaller rectangles to represent the factors.
Watch Videos:
- Watch videos that explain factoring techniques and provide examples.
- Many online resources offer video tutorials on factoring.

5.4. Collaborate with Others

Working with others can help you learn new techniques, identify your mistakes, and stay motivated.

Join a Study Group:
- Join a study group with classmates or friends who are also learning factoring.
- Work through problems together, share tips and tricks, and support each other.
Ask for Help:
- Don’t be afraid to ask for help from your teacher, tutor, or classmates.
- Getting help can clarify concepts and prevent you from making the same mistakes repeatedly.
Teach Others:
- Teaching others is a great way to reinforce your understanding of factoring.
- Explain factoring techniques to others and answer their questions.

5.5. Utilize Online Resources

Numerous online resources can help you improve your factoring skills.

Online Calculators:
- Use online calculators to check your answers and get step-by-step solutions.
- These calculators can be helpful for verifying your work and identifying mistakes.
Practice Websites:
- Use websites that offer practice problems with immediate feedback.
- These websites can help you build your skills and track your progress.
Forums and Communities:
- Join online forums and communities where you can ask questions, share tips, and connect with other learners.
- These communities can provide valuable support and guidance.

By following these tips and tricks, you can improve your factoring skills, gain confidence, and excel in your math studies.

6. Common Factoring Problems and Solutions

Factoring can be challenging, and students often encounter similar types of problems. Here are some common factoring problems and their solutions to help you navigate these challenges:

6.1. Factoring Quadratics with Leading Coefficients

Factoring quadratic expressions of the form ax^2 + bx + c where a ≠ 1 can be tricky.

Problem: Factor 2x^2 + 7x + 3
- Solution:
  - Use the “ac method.”
  - Multiply a and c: 2 × 3 = 6.
  - Find two numbers that multiply to 6 and add to 7: These are 1 and 6.
  - Rewrite the middle term: 2x^2 + x + 6x + 3.
  - Factor by grouping: x(2x + 1) + 3(2x + 1).
  - Factor out (2x + 1): (2x + 1)(x + 3).
Problem: Factor 3x^2 - 10x + 8
- Solution:
  - Use the “ac method.”
  - Multiply a and c: 3 × 8 = 24.
  - Find two numbers that multiply to 24 and add to -10: These are -4 and -6.
  - Rewrite the middle term: 3x^2 - 4x - 6x + 8.
  - Factor by grouping: x(3x - 4) - 2(3x - 4).
  - Factor out (3x - 4): (3x - 4)(x - 2).

6.2. Factoring by Grouping with More Than Four Terms

Factoring by grouping can be extended to polynomials with more than four terms by strategically grouping and factoring common factors.

Problem: Factor x^3 + 2x^2 - 3x - 6
- Solution:
  - Group terms: (x^3 + 2x^2) + (-3x - 6).
  - Factor out common factors: x^2(x + 2) - 3(x + 2).
  - Factor out (x + 2): (x + 2)(x^2 - 3).
Problem: Factor x^3 - 5x^2 + 4x - 20

Solution:
- Group terms: (x^3 - 5x^2) + (4x - 20).
- Factor out common factors: x^2(x - 5) + 4(x - 5).
- Factor out (x - 5): (x - 5)(x^2 + 4).

6.3. Factoring the Difference of Squares Multiple Times

The difference of squares pattern can be applied multiple times if the resulting factors are also differences of squares.

Problem: Factor x^4 - 16
- Solution:
  - Recognize as a difference of squares: (x^2)^2 - 4^2.
  - Factor: (x^2 + 4)(x^2 - 4).
  - Recognize that (x^2 - 4) is also a difference of squares: x^2 - 2^2.
  - Factor further: (x^2 + 4)(x + 2)(x - 2).
Problem: Factor `x^