How To Factor Third Degree Polynomial

Factoring a third-degree polynomial, also known as a cubic polynomial, might seem daunting at first, but with the right strategies and a bit of practice, it becomes a manageable task. The process involves finding the roots of the polynomial and expressing it as a product of linear and/or quadratic factors. This article will guide you through various methods, providing detailed steps and examples to help you master this skill.

Understanding Cubic Polynomials

A cubic polynomial is a polynomial of degree three. Its general form is:

ax³ + bx² + cx + d

where a, b, c, and d are constants, and a ≠ 0. Factoring such a polynomial involves finding expressions that, when multiplied together, give you the original polynomial. This often means finding the roots (the values of x for which the polynomial equals zero) and then expressing the polynomial in terms of those roots.

Methods for Factoring Cubic Polynomials

Several methods can be used to factor a cubic polynomial, including:

1. Factoring by Grouping
2. Using the Rational Root Theorem
3. Synthetic Division
4. Using Known Roots
5. Combining Methods

Let’s explore each of these methods in detail.

1. Factoring by Grouping

Factoring by grouping is a technique that can be used when the cubic polynomial has a specific structure that allows you to group terms and factor out common factors. This method is most effective when the coefficients have a particular relationship.

Steps for Factoring by Grouping:

1. Group the Terms: Arrange the polynomial into two groups of terms.
2. Factor out the Greatest Common Factor (GCF) from Each Group: Identify and factor out the GCF from each group.
3. Check for a Common Binomial Factor: If both groups now share a common binomial factor, factor it out.
4. Write the Factored Form: Express the polynomial as a product of the common binomial factor and the remaining factors.

Example:

Factor the polynomial:

x³ + 3x² - 4x - 12

1. Group the Terms:

   (x³ + 3x²) + (-4x - 12)
   

2. Factor out the GCF from Each Group:

   x²(x + 3) - 4(x + 3)
   

3. Check for a Common Binomial Factor:

   Both groups have the common binomial factor (x + 3).

4. Write the Factored Form:

   (x + 3)(x² - 4)
   

Now, notice that (x² - 4) is a difference of squares, which can be further factored:

x² - 4 = (x + 2)(x - 2)

So, the fully factored form of the polynomial is:

(x + 3)(x + 2)(x - 2)

2. Using the Rational Root Theorem

The Rational Root Theorem is a powerful tool for finding potential rational roots of a polynomial. These roots can then be used to factor the polynomial further.

The Rational Root Theorem:

For a polynomial of the form:

aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

any rational root p/q (where p and q are integers with no common factors other than 1) must satisfy:

• p is a factor of the constant term a₀.
• q is a factor of the leading coefficient aₙ.

Steps for Using the Rational Root Theorem:

1. List Possible Rational Roots: Identify all possible rational roots p/q by listing the factors of the constant term and the leading coefficient.
2. Test the Possible Roots: Use synthetic division or direct substitution to test each possible root. If the polynomial evaluates to zero for a particular root, that root is a valid root of the polynomial.
3. Use the Root to Factor: Once you find a root, use synthetic division to divide the polynomial by (x - root). This will give you a quadratic polynomial.
4. Factor the Quadratic: Factor the resulting quadratic polynomial using methods such as factoring, completing the square, or the quadratic formula.

Example:

Factor the polynomial:

2x³ - 5x² - 4x + 3

1. List Possible Rational Roots:

   • Factors of the constant term (3): ±1, ±3
   • Factors of the leading coefficient (2): ±1, ±2

   Possible rational roots: ±1, ±3, ±1/2, ±3/2

2. Test the Possible Roots:

   Let’s test x = 1:

   2(1)³ - 5(1)² - 4(1) + 3 = 2 - 5 - 4 + 3 = -4 ≠ 0
   

   So, x = 1 is not a root.

   Let’s test x = -1:

   2(-1)³ - 5(-1)² - 4(-1) + 3 = -2 - 5 + 4 + 3 = 0
   

   So, x = -1 is a root.

3. Use the Root to Factor:

   Use synthetic division to divide the polynomial by (x + 1):

   -1 |  2  -5  -4   3
      |     -2   7  -3
      ------------------
         2  -7   3   0
   

   The quotient is 2x² - 7x + 3.

4. Factor the Quadratic:

   Factor the quadratic 2x² - 7x + 3:

   2x² - 7x + 3 = (2x - 1)(x - 3)
   

So, the fully factored form of the polynomial is:

(x + 1)(2x - 1)(x - 3)

3. Synthetic Division

Synthetic division is a streamlined way to divide a polynomial by a linear factor of the form (x - c). It is particularly useful when you have identified a potential root using the Rational Root Theorem or other methods.

Steps for Synthetic Division:

1. Write Down the Coefficients: Write down the coefficients of the polynomial in order, including any zero coefficients for missing terms.
2. Set Up the Division: Write the value of c (from the factor x - c) to the left.
3. Perform the Division:
   • Bring down the first coefficient.
   • Multiply the value you brought down by c and write the result under the next coefficient.
   • Add the two values in the column.
   • Repeat the process until you have reached the last coefficient.
4. Interpret the Result: The last number in the bottom row is the remainder. The other numbers are the coefficients of the quotient polynomial.

Example:

Divide the polynomial x³ - 4x² + x + 6 by (x - 2) using synthetic division.

1. Write Down the Coefficients:

   1  -4   1   6
   

2. Set Up the Division:

   2 |  1  -4   1   6
   

3. Perform the Division:

   2 |  1  -4   1   6
      |     2  -4  -6
      ------------------
         1  -2  -3   0
   

4. Interpret the Result:

   The quotient is x² - 2x - 3, and the remainder is 0.

Therefore, x³ - 4x² + x + 6 = (x - 2)(x² - 2x - 3).

Now, factor the quadratic x² - 2x - 3:

x² - 2x - 3 = (x - 3)(x + 1)

So, the fully factored form of the polynomial is:

(x - 2)(x - 3)(x + 1)

4. Using Known Roots

If you happen to know one or more roots of the cubic polynomial, you can use this information to factor it more easily.

Steps for Using Known Roots:

1. Identify the Known Root(s): Determine the value(s) of x for which the polynomial equals zero.
2. Divide by the Corresponding Factor(s): Use synthetic division or polynomial long division to divide the polynomial by the factor (x - root) for each known root.
3. Factor the Resulting Polynomial: After dividing by the known factor(s), you will be left with a polynomial of lower degree, which is often easier to factor.

Example:

Suppose we know that x = 1 is a root of the polynomial x³ - 6x² + 11x - 6.

1. Identify the Known Root(s):

   We know that x = 1 is a root.

2. Divide by the Corresponding Factor(s):

   Divide the polynomial by (x - 1) using synthetic division:

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

   The quotient is x² - 5x + 6.

3. Factor the Resulting Polynomial:

   Factor the quadratic x² - 5x + 6:

   x² - 5x + 6 = (x - 2)(x - 3)
   

So, the fully factored form of the polynomial is:

(x - 1)(x - 2)(x - 3)

5. Combining Methods

In some cases, you might need to combine multiple methods to fully factor a cubic polynomial. For instance, you might use the Rational Root Theorem to find one root and then use synthetic division to reduce the polynomial to a quadratic, which you can then factor using traditional methods.

Example:

Factor the polynomial:

x³ - 3x² - 13x + 15

1. Use the Rational Root Theorem:

   • Factors of the constant term (15): ±1, ±3, ±5, ±15
   • Factors of the leading coefficient (1): ±1

   Possible rational roots: ±1, ±3, ±5, ±15

   Test x = 1:

   (1)³ - 3(1)² - 13(1) + 15 = 1 - 3 - 13 + 15 = 0
   

   So, x = 1 is a root.

2. Use Synthetic Division:

   Divide the polynomial by (x - 1) using synthetic division:

   1 |  1  -3  -13  15
      |     1  -2  -15
      ------------------
         1  -2  -15   0
   

   The quotient is x² - 2x - 15.

3. Factor the Quadratic:

   Factor the quadratic x² - 2x - 15:

   x² - 2x - 15 = (x - 5)(x + 3)
   

So, the fully factored form of the polynomial is:

(x - 1)(x - 5)(x + 3)

Tips and Tricks

• Look for Simple Roots First: Before diving into more complicated methods, check if simple integers like ±1, ±2, or ±3 are roots. These are often easier to test.
• Use Technology: If you are allowed, use graphing calculators or computer algebra systems (CAS) to find roots or factor polynomials. These tools can save time and reduce errors.
• Practice Regularly: Factoring polynomials becomes easier with practice. Work through a variety of examples to build your skills and intuition.
• Double-Check Your Work: After factoring a polynomial, multiply the factors together to ensure you get the original polynomial. This can help you catch any mistakes.
• Consider Complex Roots: Keep in mind that cubic polynomials can have complex roots. If you cannot find three real roots, consider the possibility of complex roots, which typically come in conjugate pairs.

Advanced Techniques

While the methods described above cover most common scenarios, there are some advanced techniques you might encounter in more complex problems.

• Cubic Formula: The cubic formula is a general formula for finding the roots of a cubic polynomial. However, it is quite complex and rarely used in practice due to its unwieldy nature.
• Numerical Methods: For polynomials with no rational roots, numerical methods such as the Newton-Raphson method can be used to approximate the roots.

Conclusion

Factoring cubic polynomials is a valuable skill in algebra, with applications in various fields, including calculus, physics, and engineering. By understanding the different methods available and practicing regularly, you can master this skill and confidently tackle a wide range of factoring problems. Whether you use factoring by grouping, the Rational Root Theorem, synthetic division, or a combination of methods, remember to approach each problem systematically and double-check your work to ensure accuracy.