Section 1: Prime Factorization Unit 11: Factoring Section 1: Prime Factorization To find the prime factorization of a whole number, it is best to start by making a factor tree Write all answers in standard form (ascending order of prime factors) Ex1. Name all of the factors of 24 Ex2. Name the first 4 multiples of 5 By the Common Factor Sum Property, if two numbers have a factor in common, then the factor will also go into their sum (and their difference)

Prime numbers are only divisible by one and the original number Composite numbers have more than 2 factors Ex3. Write the prime factorization of 630 in standard form You can use prime factorization to quickly multiply and divide large numbers Ex4. Write the prime factorization of 616 · 980 Ex5. Write the prime factorization of Section of the book to read: 12-1

Section 2: Common Monomial Factoring Common monomial factoring is the reverse of the distributive property You will be factoring out the greatest common factor from each term and showing what it looked like before the distributive property was applied Ex1. List all of the factors of 9x³ Ex2. Find the greatest common factor of and Page 767 shows you a visual way to factor

The factorization is not complete until you have factored out all of the common factors (that is why it is best to use the greatest common factor) Ex3. Factor Ex4. Factor Ex5. Factor and simplify Ex6. Factor and simplify Section of the book to read: 12-2

Section 3: Factoring x² + bx + c In Unit 2 Section 12 we learned how to FOIL when multiplying 2 binomials Ex1. Multiply (x + 4)(x – 9) In this section, we will be doing the reverse of FOIL, determining what two binomials must have been multiplied to create the given trinomial For this section only, 1 will be the only coefficient used for the x² term, so the first term in each binomial will be the variable

The two last terms will add to be b and multiply to be c Watch your signs!!! Ex2. Factor x² + 7x + 12 Ex3. Factor x² – 13x +40 Ex4. Factor x² + x – 42 If your factorization results in two identical binomials, the trinomial is called a perfect square trinomial Some trinomials cannot be factored using integers, those are said to be prime

You can factor a binomial into two binomials if it is a difference of squares (Unit 2 section 13) Ex5. Factor y² - 36 Ex6. Factor g² + 25 Sometimes you will need to perform common monomial factoring first, and then factor the quadratic that remains (ALWAYS test for this possibility first) Ex7. Factor Section of the book to read: 12-3

Section 4: Factoring ax² + bx + c Just like section 3, factoring these types of quadratics will be the reverse of FOIL There is an extra degree of difficulty with these, because now you must make sure that when you FOIL the two binomials, every term is correct You can start by guessing and checking, but your basic math skills and number sense should help you find the answer quickly First · first = ax² and last · last = c

Ex1. Factor 2x² + 13x + 15 Ex2. Factor 3x² + 11x – 4 Ex3. Factor 6x² + 7x + 2 Ex4. Factor 8x² + 2x – 15 You can use factorization to solve quadratic equations (must be = 0) Once you have factored, set each binomial = 0 and solve for the variable (next section) Section of the book to read: 12-5

Section 5: Solving Some Quadratic Equations by Factoring As discussed in the last section, sometimes you can solve quadratic equations by factoring Once factored, set each binomial (and monomial if there was first common monomial factoring) equal to 0 and solve for the variable The degree of the polynomial tells you how many potential solutions there are Ex1. Solve 3w(2w + 1)(5w – 4) = 0

In Unit 7 Section 6, we learned that the Quadratic Formula is one way to solve quadratic equations (as is factoring) The Quadratic Formula can always be used, factoring can only be used at times Solve by factoring Ex2. 6x² - 16x = 0 Ex3. 12x² + 17x = -6 Ex4. Solve 12x² + 11x – 15 = 0 Ex6. 16x³ + 32x² + 12x = 0 Section of the book to read: 12-4

Section 6: Which Quadratic Expressions are Factorable? Some quadratic expressions are not factorable using integers (they are prime) When using the quadratic formula, we found that many solutions were irrational (these would not be factorable) Using ax² + bx + c; if a, b, and c are all rational numbers and the discriminant (b² - 4ac) is a perfect square, then you can factor the expression (Discriminant Theorem)

Ex1. Is 8 + 4x² + 3x factorable? Ex2. Solve 3x² + 5x – 2 = 0 Ex3. Solve 7x² - 6x – 9 = 0 Unless otherwise directed, you may solve using the quadratic formula or by factoring Section of the book to read: 12-8