TOPIC: FACTORING POLYNOMIALS. 5.1 Introduction to Factoring I. Preliminary Thoughts A. Two concepts that get mixed up a lot are LCM and GCF

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1 MAT029A TOPIC: FACTORING POLYNOMIALS 5.1 Introduction to Factoring I. Preliminary Thoughts A. Two concepts that get mixed up a lot are LCM and GCF Note some expressions don t factor and are called prime. Least Common Multiple vs Greatest Common Factor (numbers go into it) (It goes into numbers) LCM of 4,6 is 12 GCF of 4,6 is 2 LCM of x 2, x 3 is x 3 GDF of x 2, x 3 is x 2 Often found to add fractions Often found to simplify fractions GCF is used now! B. To find the GCF with big numbers: What is the GCF of 40 and 100? Step 1: Trees Step 2: Multiply the primes they have in common. II. This first sort of factoring is Backward Distribution so check by dist n. Factor out a monomial: Step 1: Look at each coefficient to find the largest # that goes in and write it. It will be one of the numbers or something smaller. (If it s only the number one, skip it.) Step 2: If any variable shows up in each term, find the smallest exponent and write it. Step 3: Divide each term by this monomial and write the result in ( ) x 2x( - x 2) III. Factor quadri-nomials by grouping Factor t + 3 Step 1: Drop parentheses around first two terms and then second two terms. Careful of middle sign. Include it with the second grouping.

2 Step 2: Factor each set as above Step 3: Write any common binomial in parentheses. Step 4: Write leftovers in parentheses t + 3 (8 + 2 )( + 12t + 3) 2 (4t + 1) + 3(4t + 1) (4t + 1)(2 + 3) 5.2 Factoring Trinomials of the Type x 2 + bx + c I. If the coefficient of x 2 is an invisible 1, consider trial and error. If you really hate it, 5.4 provides the most general strategy for factoring trinomials so go there now. Factor + 3x + 2 Step 1: Write (x )(x ) Step 2: If sign of constant term is negative, then put different signs next. If sign of constant term is positive, then put two + s if middle term is positive OR two s if middle is negative. Step 3: Look at the constant term. If you wrote different signs, look for two factors of the last number with a difference of the middle coefficient. If you wrote same signs, look for two factors of the last number with a sum of the middle coefficient. + 3x + 2 (x )(x ) (x + )(x + ) (x + 2)(x + 1) II. General factoring tips for all factoring situations: A. Before you try anything fancy, write in descending order and look for backwards distribution, above. (factor out a -1 if it helps.) B. After you factor, check by multiplying! C. See two variables? Place combo platter in middle! 5.3 Factoring ax 2 + bx + c, a is not equal to 1: trial and error Skip this and go straight to Factoring ax 2 + bx + c, a is not equal to 1 (or not!): The ac-method I. If you want to learn just one technique beyond backwards distribution, this is it! ac/grouping Factor Step 1: Name coefficients (in descending order) a, b, c Step 2: Multiply a c Step 3: Find all pairs of factors for this. Step 4: Select pair whose sum is b. Step 5: Replace the middle term of the original trinomial using these as coefficients. (Split the middle.) Careful now you are not done yet! Step 6: Drop ( ) ( ) to start grouping as described in 5.1

3 - x 2 a1 b-1 c-2 (a)(c)(1)(-2) Factoring Trinomial squares and differences of squares. I. These are two shortcuts that depend on pattern recognition. A. Pattern recognition in binomial: Diff. of Squares Factor Step 1: Pattern Recognition: Are both terms perfect squares? (Yes) Subtraction? (Yes) Step 2: Write: ( + ( - 4 (x + 2)(x 2) Would split twice! Check it out! B. Pattern Recognition In Trinomial : Trinomial Square Factor + 6x + 9 Step 1: Pattern Recognition: Are first and last terms perfect squares? (Yes) Is the third term positive? (Yes) Is sign-less middle term 2? (yes) Step 2: Write: ( - 2 if middle is negative. ( + 2 if middle is positive. + 6x Factoring: A general strategy I. Generally it s best to have descending order and a -1 factored out if the first term is negative: (x 2-3x - 2) II. First Factoring to try is always Backwards Distribution. (5.1)

4 III. Count number of terms. A. Two terms? Try Difference of Squares shortcut (5.5) B. Four terms? Try Grouping (5.1) C. Three terms? {Note: If there are two variables, put the combo in the middle.} 1. Try Trinomial Square shortcut (5.5) 2. Try Trial and Error shortcut (5.2) 3. If these don t work, try ac/grouping (5.4) IV. See whether you can factor again (and again ) V. Check factoring by multiplying. 5.7 Solving Quadratic Equations by Factoring I. Earlier you learned how to solve linear equations ; these equations can be recognized by the fact that the unknown is not squared at any point and you usually get one solution. In quadratic equations the unknown is squared at some point and you often get two solutions. II. Steps to solve these are listed below. Easier problems don t call for all steps. You may end up jumping in later in the process! solve x (x 5) 14 A. The form you want is Factored Stuff 0. If not 1. Send all terms to one side so you have 0 on one side. 2. Factor the other side. In order to do this, you may have to first multiply and/or write terms in descending order. B. Principal of Zero: If stuff that s multiplied equals zero, then at least one factor is equal to zero. This means you should set each factor equal to zero. Check III. On any graph, x-intercepts happen when y0 so substitute 0 for y in a two variable quadratic equation to find x-intercept on parabola. 5.8 Applications of Quadratic Equations As in any word problem, Please Let Frankie Sing! (Picture/Label/Formulas/Substitutions) I. Area of a rectangle (a lw) and area of a triangle (a ½ bh) should be familiar! Look inside the cover of most texts for more formulas.

5 II. When you are given a formula, make sure you understand what variables represent and what units are to be used. III. Consecutive integers such as 2,3,4 can be represented using these expressions: x, x+1, x+2, etc. Consecutive even or odd integers need these expressions: x, x + 2, x + 4, etc. IV. Pythagorean Theorem: a 2 +b 2 c 2 if and only if you have a right triangle (Make sure c represents the hypotenuse!)