Factoring trinomials is a key skill in algebra, transforming quadratic expressions into simpler forms. This process helps solve equations and understand the behavior of quadratic functions. We'll explore methods for factoring different types of trinomials.

The AC method is a powerful tool for factoring more complex trinomials when simple inspection isn't enough. We'll also touch on additional techniques like prime factorization and synthetic division for handling trickier polynomial expressions.

Factoring Trinomials of the Form $x^2+bx+c$

Finding factors for x2+bx+c

• Find two numbers that multiply to equal the constant term $c$ and add to equal the coefficient of the $x$ term, $b$
• Rewrite the trinomial as the product of two binomials using the two numbers found $(x + \text{first number})(x + \text{second number})$
• Factor $x^2+5x+6$ by finding two numbers that multiply to 6 and add to 5 (2 and 3)
  • Rewrite the trinomial as $(x+2)(x+3)$
• This process transforms the quadratic expression into its factored form

Factoring with negative terms

• When the coefficient of the $x$ term ($b$) is negative, find two numbers that multiply to equal $c$ and add to equal the absolute value of $b$
  • One of the binomial factors will have a negative sign
  • Factor $x^2-7x+12$ by finding two numbers that multiply to 12 and add to 7 (3 and 4)
    • Rewrite the trinomial as $(x-3)(x-4)$
• When the constant term ($c$) is negative, find two numbers that multiply to equal the absolute value of $c$ and add to equal $b$
  • One number will be positive, and the other will be negative
  • Factor $x^2+2x-15$ by finding two numbers that multiply to 15 and add to 2 (5 and -3)
    • Rewrite the trinomial as $(x+5)(x-3)$

AC method for complex trinomials

• Use the AC method when factoring by inspection is not easily possible
• Steps for the AC method:
  1. Multiply $a$ (coefficient of $x^2$) and $c$ (constant term) to get $ac$
  2. Find two numbers that multiply to equal $ac$ and add to equal $b$ (coefficient of $x$)
  3. Rewrite the middle term using the two numbers found in step 2
  4. Factor by grouping
• Factor $2x^2+7x+3$ using the AC method:
  1. $ac = 2 \times 3 = 6$
  2. Find two numbers that multiply to 6 and add to 7 (1 and 6)
  3. Rewrite the middle term: $2x^2+x+6x+3$
  4. Factor by grouping: $(2x^2+x)+(6x+3) = x(2x+1)+3(2x+1) = (2x+1)(x+3)$
• The AC method relies on the distributive property to rewrite the expression

Additional Factoring Techniques

• Prime factorization can be useful when dealing with more complex polynomial expressions
• Synthetic division is an efficient method for dividing polynomials and can be used to find factors of higher-degree polynomials