Chapter+9

= 9-1 =
 * Find prime factorization of integers and monomials.
 * Find the greatest common factors of integers and monomials.

Prime number- it's only factor is itself and 1 . Composite number- more than 2 factors . GCF- **g** reatest **c** ommon **f** actor . Factored form- All numbers are factored and NO exponents



Example: 12x^2yz^2 : 2 * 2 * 3 * x * x * y * z * z * z  ... 18xyz^2 : 2 * 3 * 3 * x * y * x * z * z. Prime factorization of a NEGATIVE number: add -1 into the factorization

= 9-2 =
 * Factor polynomials by using the Distributive Property.
 * Solve quadratic equations of the form ax^2 + bx = 0.

Examples: //Easy// __2a + 6__ 2a: 2 * a  .. 6: 2 * 3

//2(a + 3)//

//Medium// __x + x^2y + x^3 + y^2__ x x + x + y  x + x + x + y + y

x (1 + xy + x^2y^2)

//Hard// 8p^2q^2 - 24pq^2 + 16 pq

8pq (pq - 3q^2 + 2)

__Steps:__

1. Break into sets of terms

2. Do the distributive property thing

3. Outside of parentheses together

4. Inside parentheses (same things) after

= 9-3 =
 * Factor trinomials of the form x^2 + bx + c.
 * Solve equations of the form x^2 + bx + c = 0.

Factoring Trinomials x^2 + bx + c


 * //Look for two numbers that multiply to equal c but add to equal b// **

__ Example __ x^2 + 7x + 10 (x + 5) (x + 2) (-5, -2)

Both numbers are positive = ( + ) ( + ) Negative inside and positive end = ( - ) ( - )

Solution Set- the number plus itself will equal 0 ................... {-10, -2}  ................... a + 10 = 0 ................... a = -10

= 9-4 =
 * Factor trinomials of the form ax^2 + bx + c.
 * Solve equations fo the form ax^2 + bx + c = 0.

ax^2 + bx + c // ac = mulyiply // // .. b= add // //.//

**Prime equation = no common factors to add/multiply** **.** Steps **.** **.** = 9.5 =
 * 1) ====** Look for a greatest common factor (if their is one) **====
 * 2) Multiply a and c (that is what you multiply to; b is what you add to)
 * 3) Put solution set back into the problem ............................................................. //If their is no set possible, it is prime!//
 * 4) Factor by grouping
 * 5) Put the =0 in
 * 6) Two step equation
 * 7) variable= .. //or// variable=
 * 8) { .., ... }
 * 9) Check you work (plug solution set back into the problem)
 * Combine terms to only have 3 terms per equation**
 * Factor binomials that are the differences of squares.
 * Solve equations involving the differences of squares.

You are not done untill the x's dont have variables

Easy a^2 - 16 (a + 4) (a - 4) a^2 - 4a + 4a - 16

Hard -2 + 2x^2 2x^2 - 2 2 (x^2 - 1) 2

Harder 1/4x^2 - 25 = 0 (1/2 + 5) (1/2 - 5) = 0 1/2 x + .5 = 0 ......... 1/2x - 5 = 0 1/2x = -5 ................ 1/2x = 5 x = -10 ................... x = 10 .............. {-10, 10}

Prime 72x^2 - 32 = . = = 9.6 =
 * Factor perfect square trinomials.
 * Solve equations involving perfect squares.