Factored form will always look like:
$$y=(xpm h)(xpm k)$$
The key thing to remember with factoring is that when you add (h+k), it must equal the coefficient to x (–2 in the equation above). When you multiply (htimes k), it must equal the constant (–3 in the equation above). Therefore, h and k must equal (–3) and (1). So our equation in factored form would be:
$$y=(x-3)(x+1)$$
If we drew a graph of (y=(x-3)(x+1)), it would be exactly the same as the graph of (y=2x^2-4x-6) pictured before. The difference in the two equations is that we can easily figure out the x-intercepts if the equation is in factored form. The x-intercepts are the x coordinates when the y value is 0. Looking at the factored equation above, we can see that y will be 0 if (x = 3) or (x = -1). That means that the graph crosses over the x-axis at those two points.
Let’s now go over some common factoring techniques that will help us break down complex expressions into simple components (factors) that will help us identify x-intercepts of equations.
Common Factoring Techniques
- Factoring Out the Greatest Common Factor (GCF):
- Identify the greatest common factor shared by the terms in the expression.
- Example:
- For (6x^2+12x) the GCF is (6x). Factored form: (6x(x+2))
- Factoring Trinomials:
- Commonly used for quadratic expressions of the form (ax^2+bx+c).
- Look for two numbers that multiply to ac and add up to b.
- Example:
- (x^2+5x+6) factors to ((x+2)(x+3))
- Difference of Squares:
- Applies to expressions like (a^2-b^2).
- The factored form will always be: ((a+b)(a-b))
- Example:
- (x^2-9) factors to ((x+3)(x-3))
- Perfect Square Trinomials:
- Recognize patterns like (a^2+2ab+b^2) or (a^2-2ab+b^2).
- Factored form: ((a+b)^2) or ((a-b)^2)
- Example
- (x^2+6x+9) factors to ((x+3)^2)
- Factoring by Grouping:
- Used when an expression has four or more terms.
- Group terms to factor out common elements, then factor the entire expression.
- Example:
- (x^3+3x^2+2x+6) can be factored as (x^2(x+3)+2(x+3)), and then ((x^2+2)(x+3))
Tips for Effective Factoring:
- Always start by factoring out the GCF if possible.
- Check for special patterns (difference of squares, perfect square trinomials).
- Practice recognizing patterns quickly; speed is essential in standardized tests.
- Review and practice with different types of factoring problems.
Practice:
Practice factoring the following equations:
(y=x^2+8x+15)
(y=x^2+x-12)
(y=x^2-5x+6)