How to find a polynomial in factored form?

To find a polynomial in factored form, you transform it from a sum of terms into a product of simpler polynomials, often binomials or trinomials, that cannot be factored further. This process is called factoring.

Understanding Factored Form

A polynomial in factored form is expressed as a multiplication of its factors. For example, the polynomial x² + 5x + 6 in standard form can be written as (x + 2)(x + 3) in factored form. This form is particularly useful for finding the roots of a polynomial, simplifying expressions, and solving equations.

Key Factoring Techniques

Various methods are employed to factor polynomials, depending on the number of terms and their structure. Always begin by looking for a Greatest Common Factor (GCF) as the first step in any factoring problem.

1. Factoring out the Greatest Common Factor (GCF)

The GCF is the largest term (number, variable, or both) that divides evenly into all terms of the polynomial.

• Process: Identify the GCF among all terms and divide each term by it, placing the GCF outside parentheses.
• Example:
  • 6x³ - 9x² + 12x
  • The GCF is 3x.
  • Factored Form: 3x(2x² - 3x + 4)

2. Factoring by Grouping

This method is highly effective for polynomials with four terms. It involves organizing terms into pairs and then factoring out common factors from each pair.

• Step 1: Group Terms. Arrange the four terms into two groups, typically the first two and the last two. For instance, (ax³ + bx²) + (cx + d).
• Step 2: Factor GCF from Each Group. Find the greatest common factor within each of the two binomial groups you created and factor it out separately. This should result in a common binomial factor appearing in both groups. For example, x²(ax + b) + c(ax + b).
• Step 3: Factor out the Common Binomial. If a common binomial factor is present in both resulting terms, factor that entire binomial out. The final result will be a product of two binomials. This looks like (ax + b)(x² + c).

To ensure accuracy, you can always multiply your answer back out to check if it matches the original polynomial. For a detailed guide on this method, refer to Factoring Polynomials with Grouping.

• Example:
  • x³ + 2x² + 3x + 6
  • Step 1 (Group): (x³ + 2x²) + (3x + 6)
  • Step 2 (Factor GCF from each group): x²(x + 2) + 3(x + 2)
  • Step 3 (Factor common binomial): (x + 2)(x² + 3)
  • Factored Form: (x + 2)(x² + 3)

3. Factoring Trinomials (Quadratic Form: ax² + bx + c)

Trinomials are polynomials with three terms. The approach depends on the leading coefficient (a).

a. When the Leading Coefficient is 1 (x² + bx + c)

• Process: Look for two numbers that multiply to the constant term (c) and add up to the middle term's coefficient (b).
• Example:
  • x² + 5x + 6
  • Numbers that multiply to 6 and add to 5 are 2 and 3.
  • Factored Form: (x + 2)(x + 3)

b. When the Leading Coefficient is Not 1 (ax² + bx + c, where a ≠ 1)

• Process (AC Method):
  1. Multiply a and c.
  2. Find two numbers that multiply to ac and add up to b.
  3. Rewrite the middle term (bx) using these two numbers.
  4. Factor the resulting four-term polynomial by grouping (as described above).
• Example:
  • 2x² + 7x + 3
  • a = 2, b = 7, c = 3. ac = 2 * 3 = 6.
  • Numbers that multiply to 6 and add to 7 are 1 and 6.
  • Rewrite 7x as x + 6x: 2x² + x + 6x + 3
  • Factor by grouping: x(2x + 1) + 3(2x + 1)
  • Factored Form: (2x + 1)(x + 3)

4. Special Factoring Patterns

Recognizing these patterns can significantly speed up the factoring process.

By systematically applying these techniques, starting with the GCF, you can efficiently break down polynomials into their factored forms.