Name Intro to Algebra 2. Unit 1: Polynomials and Factoring


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1 Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332 #3, 4, 5, 47, 48, 60 9/8 4 Multiplying Polynomials Pg. 332 #36, 40, 45, 46 9/9 5/6 Binomial Expansion & Pascal s Triangle None 9/10 7 Pascal s Triangle with substitutions Pg 1068 #1, 2, 6, 9, 11, 15 9/11 8 QUIZ Factoring; GCF and Difference of Perfect Squares Pg. 337#1, 2, 3, 4, 7, 8, 9 9/12 9 Factoring; FOIL Backwards Pg. 337 # 10,11, 13,14,15 9/15 10/11 Factoring by Grouping Worksheet 9/16 12/13 Quiz Factoring trinomials with leading coefficient 1 Pg # 12, 18, 21 9/17 x More Practice No Homework 9/18 14 Factoring completely Pg 338 # 22, 26, 30, 31, 41 9/19 Review Finish Review 9/22 Review Study 9/23 Test No Homework 1
2 Polynomial Vocabulary Term Definition Examples Polynomial Monomial Binomial Trinomial Degree of a monomial Degree of a polynomial Standard Form Linear Polynomial Quadratic Polynomial Cubic Polynomial Find the degree of the following: Name the following polynomials (by # of terms and degree): 2
3 Adding and Subtracting Polynomials When adding or subtracting polynomials, you may only COMBINE LIKE TERMS! Perform the indicated operations: 1.)(2x + 3)  (x  4) + (x + 2) 2.) (x 2 + 4)  (x  4) + (x 22x) 3.)(3x 25x)  (x 2 + 4x + 3) 4.) 2x(3x 2  x)  (x 2 + 2x + 7) 5.) (7x  4x ) + 3(7x 2 + 5) 6.) (4x + 7x 39x 2 ) + x 2 (32x 25x) 7.) (y3 + y 22) + (y  6y 2 ) 8.) (x 28x  3)  (x 3 + 8x 28) 9.) (3x 22x + 9)  (x 2  x + 7) 10.) (2x 26x + 3)  (2x + 4x 2 + 2) 3
4 Multiplying Polynomials When multiplying a binomial by a binomial you use FOIL. Or just make sure each term in the first set of parenthesis is multiplied by each term in the second set. 1.) 4(a  3) 2.) 5(x  2) 3.) 3x 2 (x 2 + 3x) 4.) 4x 3 (x  3) 5.) 5x 2 (x 2 + 2x + 1) 6.) (x+4)(x2) 7.) (x 2 +3)(x1) 8.) (x+4)(x 2 +5x+7) 9.) (2x6)(x 27x2) **10.) (x 2 +3x2)(x 25x+1) 4
5 BINOMIAL EXPANSION Compute the following by multiplying out the binomials: 1.) (x+y) 2 2.) (x+y) 3 3.) Consider the following binomial expansion: (x+y) 7, write it out and think about how you would multiply out each binomial Notice the pattern in Pascal s triangle and fill in the remaining boxes: 5
6 USING PASCAL S TRIANGLE Write the expansion of the given binomial: 1.) (x+y) 3 2.) (x+y) 5 Now let s go back and do the expansion from the previous page: 3.) (x+y) 7 6
7 Pascal s Triangle with Substitutions Write the expansion of the given binomial by substituting for x and y. 3.) (x+4) 4 4.) (a2) 6 WRITING A TERM OF AN EXPANSION Find the given term of the binomial expansion below: 1.) (k+3) 5, 3 rd term: 2.) (2x+1) 7, 6 th term: 3.) (2a3) 5, 4 th term: 7
8 Factoring by GCF and Difference of Perfect Squares FACTORING USING GCF. 1.) Look for a common factor in each term. 2.) Write the common factor outside of a set of parenthesis. 3.) Write what's left inside the parenthesis. 1.) 3x+6 2.) 2x 2 +4x 3.) 16x 3 y32xy 2 4.)25x 5 y 2 z+15x 2 yz 35xyz 2 5.) 14x 2 y 8 z 3 +24xy 4 z 22xyz FACTORING USING DIFFERENCE OF PERFECT SQUARES: 1.) Check to see if it has ONLY TWO TERMS and that each one is a PERFECT SQUARE and that there is a minus in the middle. 2.) Write two set of parenthesis. 3.) In the First Spot in each set, write the square root of the first term of your binomial 4.) In the Second Spot in each set, write the square root of the second term of your binomial 5.) Put a + in one set of () and a in the other. 6.) x ) 4x ) 16x 2 9.) x
9 FACTORING USING FOIL BACKWARDS: 1.) Check to see if it is a trinomial. 2.) Write two sets of parenthesis. In the first spot put factors of the first term in the trinomial 3.) in the second spot put factors of the third term in the trinomial. a.) If the last term is +, the factors have to add to the middle term and the signs will both have the same sign as the middle term. b.) If the last term is , the factors have to subtract to the middle term and the signs will be different, the bigger # gets the sign of the middle term. Factor each trinomial: 1.)x 2 +10x+9 2.) x 212x+27 3.) x 2 +5x14 4.) x 213x+40 5.) x 23x ) m 2 + 4m ) x 22x ) x 211x ) b 2 + 4b ) x 2 + 9x
10 FACTORING BY GROUPING 1.) Factor x 3 + 2x 2 + 8x + 16 Ask yourself: Are there common factors to all four terms? Is it a trinomial? Can we use the quadratic formula? *Looks like we need a new method to solve this* 2.) Factor xy 4y + 3x 12 3.) Factor xy 4y 3x
11 PRACTICE. Factor each of the following by grouping: 1. xy + 7x + 4y xy + 5x + 10y + 25 = x( ) + 4( ) = ( )( ) 3. x 3 + 3x 2 + 9x x 3 3x 2 + 9x ax + bx + ay + by 6. ax + ac + bx + bc 7. ax bx + ay by 8. ax ac + bx bc 11
12 FACTORING A TRINOMIAL with LEADING COEFFICIENT 1 We use factoring by grouping! Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem is in the correct order. Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the three terms only have a 1 in common which is of no help. Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 6 and 2. Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 3 and 4 can combine to equal 1. Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, 3 and +4 combine to equal +1 and 3 times +4 is 12. Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5. Step 7: Now that the problem is written with four terms, you can factor by grouping. 12
13 Now you try! Example 2 Factor: Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem is in the correct order. Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the three terms only have a 1 in common which is of no help. Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 12 and 15. Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 9 and 20 can combine to equal 29. Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, 9 and 20 combine to equal 29 and 9 times 20 is 180. Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5. Step 7: Now that the problem is written with four terms, you can factor by grouping. 13
14 FACTOR COMPLETELY: 1.) Always look for a GCF first! 2.) Then look to see if you can use Difference of Perfect Squares. 3.) Finally, factor using FOIL Backwards if possible 1.) x ) 2x ) 8x 340x 2 +48x 4.) 2x 2 + 4x ) 8x 332x 6.)4x 232x ) 14
15 15
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