How To Factorise A Polynomial Of Degree 3

Factoring a polynomial of degree 3, also known as a cubic polynomial, involves expressing it as a product of simpler polynomials, ideally linear and quadratic factors. This process is fundamental in algebra and calculus, simplifying complex expressions and solving equations.

Understanding Cubic Polynomials

A cubic polynomial is generally represented as:

ax³ + bx² + cx + d

where a, b, c, and d are constants and a ≠ 0. Factoring such polynomials can be challenging but achievable through several methods.

Methods for Factoring Cubic Polynomials

Several techniques can be employed to factorize a cubic polynomial:

• Rational Root Theorem: This theorem helps identify potential rational roots of the polynomial.
• Factor Theorem: Once a root is found, the Factor Theorem allows us to extract a corresponding linear factor.
• Synthetic Division: An efficient method for dividing the polynomial by a linear factor.
• Factoring by Grouping: Applicable when the polynomial can be grouped into pairs with common factors.

Let's explore each method in detail.

1. Rational Root Theorem

The Rational Root Theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term (d) and q is a factor of the leading coefficient (a).

Steps:

1. Identify Possible Rational Roots: List all possible values of p/q. This involves finding all factors of d and a, and then forming all possible fractions.
2. Test Possible Roots: Substitute each possible root into the polynomial. If the result is zero, then that value is a root of the polynomial.

Example:

Consider the polynomial:

f(x) = x³ - 6x² + 11x - 6

1. Identify Possible Rational Roots:
   • Factors of d (-6): ±1, ±2, ±3, ±6
   • Factors of a (1): ±1
   • Possible rational roots: ±1, ±2, ±3, ±6
2. Test Possible Roots:
   • f(1) = (1)³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0. Therefore, x = 1 is a root.

2. Factor Theorem

The Factor Theorem states that if f(a) = 0 for a polynomial f(x), then (x - a) is a factor of f(x).

Steps:

1. Find a Root: Use the Rational Root Theorem or trial and error to find a root a of the polynomial.
2. Apply Factor Theorem: If a is a root, then (x - a) is a factor of the polynomial.

Example (Continuing from above):

Since we found that x = 1 is a root of f(x) = x³ - 6x² + 11x - 6, then (x - 1) is a factor.

3. Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear factor. It is particularly useful when combined with the Factor Theorem.

Steps:

1. Set Up: Write down the coefficients of the polynomial and the root a that you are dividing by.
2. Perform Division:
   • Bring down the first coefficient.
   • Multiply the root by this coefficient and write the result under the next coefficient.
   • Add these two numbers.
   • Repeat the process until all coefficients have been used.
3. Interpret Results: The last number is the remainder, and the other numbers are the coefficients of the quotient polynomial.

Example (Continuing from above):

Divide x³ - 6x² + 11x - 6 by (x - 1) using synthetic division.

1 |  1  -6  11  -6
  |      1  -5   6
  ------------------
    1  -5   6   0

The quotient is x² - 5x + 6, and the remainder is 0.

4. Factoring by Grouping

Factoring by grouping is effective when the cubic polynomial can be arranged into pairs with common factors.

Steps:

1. Group Terms: Arrange the polynomial into two groups of terms.
2. Factor Each Group: Factor out the greatest common factor (GCF) from each group.
3. Identify Common Binomial Factor: If both groups share a common binomial factor, factor it out.

Example:

Consider the polynomial:

f(x) = x³ - 2x² - 4x + 8

1. Group Terms: (x³ - 2x²) + (-4x + 8)
2. Factor Each Group: x²(x - 2) - 4(x - 2)
3. Identify Common Binomial Factor: (x - 2)(x² - 4)
4. Further Factorization (if possible): (x - 2)(x - 2)(x + 2) = (x - 2)²(x + 2)

Complete Example: Factoring f(x) = x³ - 6x² + 11x - 6

1. Rational Root Theorem:
   • Possible rational roots: ±1, ±2, ±3, ±6
   • Testing x = 1: f(1) = 1³ - 6(1)² + 11(1) - 6 = 0. So, x = 1 is a root.
2. Factor Theorem:
   • Since x = 1 is a root, (x - 1) is a factor.
3. Synthetic Division:
   • Dividing x³ - 6x² + 11x - 6 by (x - 1) gives a quotient of x² - 5x + 6.
4. Factoring the Quotient:
   • Factor x² - 5x + 6 as (x - 2)(x - 3).
5. Complete Factorization:
   • f(x) = (x - 1)(x - 2)(x - 3)

Special Cases and Advanced Techniques

While the above methods cover many cubic polynomials, some special cases and advanced techniques may be needed:

• Perfect Cubes: Polynomials of the form a³ ± 3a²b + 3ab² ± b³ can be factored directly as (a ± b)³.
• Sum/Difference of Cubes: Polynomials of the form a³ ± b³ can be factored as (a ± b)(a² ∓ ab + b²).
• Numerical Methods: When the polynomial has irrational or complex roots, numerical methods like Newton-Raphson iteration can be used to approximate the roots.
• Complex Numbers: Sometimes, the roots of a cubic polynomial can be complex numbers. In such cases, the factorization will involve complex conjugate pairs.

Dealing with Irreducible Quadratics

After extracting a linear factor, you may be left with an irreducible quadratic, meaning it cannot be factored further using real numbers. This occurs when the discriminant (b² - 4ac) of the quadratic is negative.

Example:

Suppose we have factored a cubic polynomial to obtain (x - 2)(x² + x + 1). The quadratic x² + x + 1 has a discriminant of 1² - 4(1)(1) = -3, which is negative. Therefore, x² + x + 1 is irreducible over real numbers.

Tips and Tricks

• Look for Simple Roots: Start by testing simple integer values like ±1, ±2, etc., as these are often roots of the polynomial.
• Use Technology: Tools like graphing calculators or computer algebra systems (CAS) can help find roots and perform polynomial division.
• Practice: Factoring polynomials requires practice. Work through various examples to become comfortable with the different techniques.
• Be Organized: Keep your work organized to avoid mistakes, especially when using synthetic division or factoring by grouping.
• Double-Check: After factoring, multiply the factors back together to ensure they match the original polynomial.

Practical Applications

Factoring cubic polynomials is not just an algebraic exercise; it has practical applications in various fields:

• Engineering: Solving for roots of polynomials is essential in control systems, signal processing, and structural analysis.
• Physics: Polynomials appear in equations describing motion, energy, and other physical phenomena.
• Computer Graphics: Polynomials are used to model curves and surfaces in computer graphics and animation.
• Economics: Polynomials can model cost functions, revenue functions, and other economic relationships.

Example Problems

Here are some additional examples to illustrate the process of factoring cubic polynomials:

Example 1:

Factor f(x) = x³ + 6x² + 11x + 6.

1. Rational Root Theorem: Possible roots: ±1, ±2, ±3, ±6.
   • f(-1) = (-1)³ + 6(-1)² + 11(-1) + 6 = -1 + 6 - 11 + 6 = 0. So, x = -1 is a root.
2. Factor Theorem: (x + 1) is a factor.
3. Synthetic Division:

-1 |  1   6  11   6
   |     -1  -5  -6
   ----------------
     1   5   6   0

The quotient is x² + 5x + 6.

4. Factoring the Quotient: x² + 5x + 6 = (x + 2)(x + 3).
5. Complete Factorization: f(x) = (x + 1)(x + 2)(x + 3).

Example 2:

Factor f(x) = 2x³ - 5x² + 4x - 1.

1. Rational Root Theorem: Possible roots: ±1, ±1/2.
   • f(1) = 2(1)³ - 5(1)² + 4(1) - 1 = 2 - 5 + 4 - 1 = 0. So, x = 1 is a root.
2. Factor Theorem: (x - 1) is a factor.
3. Synthetic Division:

1 |  2  -5   4  -1
  |      2  -3   1
  ----------------
    2  -3   1   0

The quotient is 2x² - 3x + 1.

4. Factoring the Quotient: 2x² - 3x + 1 = (2x - 1)(x - 1).
5. Complete Factorization: f(x) = (x - 1)(2x - 1)(x - 1) = (x - 1)²(2x - 1).

Example 3:

Factor f(x) = x³ - 7x - 6.

1. Rational Root Theorem: Possible roots: ±1, ±2, ±3, ±6.
   • f(-1) = (-1)³ - 7(-1) - 6 = -1 + 7 - 6 = 0. So, x = -1 is a root.
2. Factor Theorem: (x + 1) is a factor.
3. Synthetic Division:

-1 |  1   0  -7  -6
   |     -1   1   6
   ----------------
     1  -1  -6   0

The quotient is x² - x - 6.

4. Factoring the Quotient: x² - x - 6 = (x - 3)(x + 2).
5. Complete Factorization: f(x) = (x + 1)(x - 3)(x + 2).

Conclusion

Factoring cubic polynomials is a valuable skill in algebra with numerous applications. By mastering the Rational Root Theorem, Factor Theorem, Synthetic Division, and Factoring by Grouping, you can efficiently break down complex cubic expressions into simpler factors. While some polynomials may require more advanced techniques or numerical methods, the fundamental principles remain the same. Practice and familiarity with these methods will enhance your ability to tackle a wide range of polynomial factorization problems.