1. FACTORING

2.  To find the prime factorization of a whole number, make a factor tree and factor until all of the factors are prime  Write the final answer in standard form  The factors will be in ascending order  Ex1. Name all of the factors of 24  Ex2. Name the first four multiples of 5  If two numbers have a factor in common, then the factor will also go into their sum (and their difference)  See the Common Factor Sum Property  i.e. if 3 goes in to 6 and 15, then 3 goes into 21 and 9

3.  Ex3. Write the prime factorization of 630 in standard form  You can use prime factorization to multiply and divide large numbers without a calculator  Find the prime factorization and then follow the rules for exponents  Ex4. Write the prime factorization of 616 · 980  Ex5. Write the prime factorization of and then simplify  Section of the book to read: 12-1

4.  Common monomial factoring is the reverse of the distributive property  Factor (or divide) out the largest common factor  Ex1. List all of the factors of 9x²  Ex2. Find the greatest common factor of and  Open your book to page 727 so we can see the visual way to factor  The factorization is only complete when there are no more common factors for the terms within the parentheses

5.  Factor completely  Ex3.  Ex4.  You can use factoring to simplify  Either factor then simplify or simplify then factor  Factor and simplify  Ex5.  Ex6.  Section of the book to read: 12-2

6. WWhen factoring quadratics, always first check to see if it is possible to do any common monomial factoring NNext, reverse FOIL to see what two binomials multiply together to make the given quadratic IIf the leading coefficient (the coefficient to the x² term) is one: TThe two last terms must multiply to be c TThe two last terms must add to be b WWatch your signs!!! EEx1. Name two numbers that multiply to be 20 and add to be 9

7. EEx2. Factor x² + 9x + 20 EEx3. Factor x² + 7x + 12 EEx4. Factor x² ─ 13x + 40 EEx5. Factor x² + x ─ 42 IIn a later section we will learn how to factor quadratics that have other leading coefficients IIf your factoring results in two identical binomials, then the trinomial is called a perfect square trinomial EEx6. Factor x² + 10x + 25 SSo x² + 10x + 25 is a perfect square trinomial IIf a trinomial cannot be factored, it is said to be prime TThe answer would be “prime”

8. BBinomials that are a difference of squares (perfect square minus perfect square) are also factorable TThe two parentheses will be the same except one will be addition and one will be subtraction EEx7. Factor y² ─ 36 EEx8. Factor g² + 25 EEx9. Factor SSection of the book to read: 12-3

9. TThe quadratic trinomials will be factored first by common monomial factoring (if possible) and then by reverse FOIL TThe two first terms must multiply to be ax² TThe two last terms must multiply to be c TThe product of the outer terms plus the product of the inner terms must be bx TThis will require some amount of guess and check GGood number sense will speed up this process! AAlways check your work by FOILing it back out

10. FFactor each of the following EEx1. 2x² + 13x + 15 EEx2. 3x² + 11x – 4 EEx3. 6x² + 7x + 2 EEx4. 8x² + 2x – 15 SSection of the book to read: 12-5

11.  Some quadratics can be solved by factoring  This is an alternative to the quadratic formula  The equation must be set = 0  The quadratic formula can ALWAYS be used, but factoring does not always work  Once the expression is factored, set each factor = 0 and solve for the variable  The degree of the polynomial will be the number of solutions  Ex1. Solve 5x(2x + 1)(x – 8) = 0

12. SSolve by factoring EEx2. 6x² - 16x = 0 EEx3. 12x² + 17x = -6 EEx4. 12x² + 11x – 15 = 0 EEx5. 16x³ + 32x² + 12x = 0 SSection of the book to read: 12-4

13.  Some quadratic expressions are not factorable using integers because they are prime  By the Discriminant Theorem, if the Discriminant (b² - 4ac) is a perfect square, then the quadratic is factorable  So find the Discriminant first to see if you should try to factor the quadratic  Are the following quadratics factorable? If so, factor it.  Ex1. 4x² + 5x – 3  Ex2. 6x² + 13x – 5  Section of the book to read: 12-8