Factoring Polynomials

Introduction to Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler expressions that, when multiplied together, give the original polynomial. Factoring helps in solving polynomial equations and simplifying expressions.

Common Factoring Techniques

1. Finding the Greatest Common Factor (GCF)

The GCF is the largest factor that divides each term in the polynomial. To factor using the GCF:

• Identify the GCF of all terms.
• Factor out the GCF from each term.

Example: Factor 6x² + 9x.
GCF: 3x → 3x(2x + 3)

2. Factoring by Grouping

Used when a polynomial has four terms. Steps include:

• Group terms into two pairs.
• Factor out the common factor in each group.
• Factor out the common binomial.

Example: Factor x³ + 3x² + 2x + 6.
Grouping: (x³ + 3x²) + (2x + 6)
Factor: x²(x + 3) + 2(x + 3)
Final Factorization: (x² + 2)(x + 3)

3. Factoring Trinomials

Trinomials in the form of ax² + bx + c can be factored by finding two numbers that multiply to ac and sum to b.

Example: Factor x² + 5x + 6.
Find two numbers that multiply to 6 and add to 5 (2 and 3).
Factor: (x + 2)(x + 3)

4. Factoring the Difference of Squares

If a polynomial is in the form a² - b², it factors into (a - b)(a + b).

Example: Factor x² - 16.
Recognize as (x)² - (4)².
Factor: (x - 4)(x + 4)