1. Binomials

2. What is a binomial?  A binomial expression is an expression with 2 terms.  EXAMPLES: x+2, 2p-3, p+q

3. Binomial Products  The product of two binomial expressions is called a binomial product.  EXAMPLES: (x+2)(x-3); (2p-3)(p-2)

4. How do we solve binomial products?  Remember when we have 2 binomial expressions we can solve them using FOIL First Outer Inner Last  (x+2)(x-3)= (x+2)(x-3) first, (x+2)(x-3) outer (x+2)(x-3) inner (x+2)(x-3) last

5. Examples!  (x+4)(2x-3) 2x 2 +5x-12  (p+q)(p-q) p 2 -q 2  (3r+2)(r+5) 3r 2 +17r+10  (t-6)(t-3) t 2 -9t+18  (x-4)(x+4) X 2 -16  (4x-2)(x+3) 4x 2 +10x-6  (2t+1)(t-3) 2t 2 -5t-3

6. Factoring Quadratics  A "quadratic" is a polynomial that looks like "ax2 + bx + c", where "a", "b", and "c" are just numbers. For the easy case of factoring, you will find two numbers that will not only multiply to equal the constant term "c", but also add up to equal "b", the coefficient on the x-term.

7. How to factor  For instance, x 2 + 5x + 6. We need to look at the factors of 6 and see which add up to 5 since both signs are addition.  Since 2(3)=6 and 2+3=5, then 2 and 3 must be the factors.  Therefore, x 2 + 5x + 6=(x+2)(x+3)

8. Now you try….  x 2 – 5x + 6 (x-2)(x-3)  x 2 + 7x + 6 (x+6)(x+1)  x 2 – 7x + 6 (x-6)(x-1)  x 2 + x – 6 (x+3)(x-2)  x 2 – x – 6 (x-3)(x+2)  x 2 + 7x – 6 unfactorable

9. Different Cases  There may be cases where the coefficient in front of the squared term is not 1. Then we need to look at each term separately.

10. For instance….  Take the example 2x 2 + x – 6  2x 2 + x – 6 =  2x 2 + 4x – 3x – 6 =  2x(x + 2) – 3(x + 2) =  (x + 2)(2x – 3) Find the value of ac to help you figure out what terms to use for the middle term! Since ac=-12, look at the factors of 12 and see what subtracts to 1.

11. Try it!  4x 2 – 19x + 12  4x 2 -16x-3x+12  4x(x-4)-3(x-4)  (4x-3)(x-4)  5x 2 – 10x + 6  Since ac=30, we look at the factors of 30: 30 and 1, 3 and 10, 5 and 6, 2 and 15. But none of these add up to 10, therefore this is unfactorable.