How did you factor polynomials?

I factor polynomials using a variety of techniques, depending on the complexity of the polynomial. Here's a breakdown of the methods I use:

• Identifying Common Factors: The first step is always to look for a common factor in all terms of the polynomial. I can then factor this out. For example, in the expression 3x^2 + 6x, I identify that 3x is a common factor, so I factor it out to get 3x(x + 2).

• Factoring by Grouping: This method is particularly useful for polynomials with four or more terms. I group the terms into pairs or groups, and then factor out the greatest common factor (GCF) from each group. If the remaining expressions in the parentheses are identical, I can factor them out as well. As per the reference, for example, with x^3 + x^2 – x – 1, I would group it as (x^3 + x^2) + (-x - 1). From the first group, I factor out x^2 to get x^2(x + 1). From the second group, I factor out -1 to get -1(x + 1). Now, I have x^2(x + 1) - 1(x + 1), and since (x + 1) is common, I factor it out to get (x + 1)(x^2 - 1). Finally, I can further factor (x^2 - 1) using the difference of squares.

• Factoring Quadratic Trinomials (ax² + bx + c): This is a common type of factoring.

  • When a = 1: I look for two numbers that multiply to c and add up to b. For example, to factor x^2 + 5x + 6, I need two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3, so the factored form is (x + 2)(x + 3).
  • When a ≠ 1: This is a bit more complex. I can use the "ac method" or the "grouping method". For example, to factor 2x^2 + 5x + 2, I multiply a (2) and c (2) to get 4. Then, I look for two numbers that multiply to 4 and add up to b (5). These numbers are 1 and 4. Then I rewrite the middle term: 2x^2 + x + 4x + 2. Now factor by grouping: x(2x + 1) + 2(2x + 1), which simplifies to (2x + 1)(x + 2).

• Difference of Squares (a² - b²): I recognize this pattern and factor it as (a + b)(a - b). For example, x^2 - 9 factors to (x + 3)(x - 3).

• Perfect Square Trinomials (a² + 2ab + b² or a² - 2ab + b²): I recognize these patterns and factor them as (a + b)² or (a - b)², respectively. For example, x^2 + 6x + 9 factors to (x + 3)².

• Sum and Difference of Cubes (a³ + b³ or a³ - b³): I use the following formulas:

  • a³ + b³ = (a + b)(a² - ab + b²)
  • a³ - b³ = (a - b)(a² + ab + b²)
    For example, x^3 + 8 factors to (x + 2)(x^2 - 2x + 4).

• Using the Rational Root Theorem: If I'm dealing with a higher-degree polynomial and cannot easily identify factors, I can use the Rational Root Theorem to find potential rational roots. This helps in finding linear factors.

• Polynomial Long Division or Synthetic Division: Once I find a root (e.g., using the Rational Root Theorem), I can use polynomial long division or synthetic division to divide the polynomial by the corresponding linear factor. This reduces the degree of the polynomial and makes it easier to factor further.

After applying these methods, I always double-check my factored form by expanding it to ensure it matches the original polynomial.