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M3U3D1 Warm-UP Distribute each problem:

M3U3D1 Warm-UP Distribute each problem:. COLLECT WARMUPS. Evaluating and Operating with Polynomials OBJ: To review adding, subtracting, multiplying, and factoring polynomials. How do I evaluate polynomial functions? You have 3 minutes to complete the top of handout page 1. Discuss.

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M3U3D1 Warm-UP Distribute each problem:

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  1. M3U3D1 Warm-UP Distribute each problem: COLLECT WARMUPS

  2. Evaluating and Operating with Polynomials OBJ: To review adding, subtracting, multiplying, and factoring polynomials

  3. How do I evaluate polynomial functions? You have 3 minutes to complete the top of handout page 1. Discuss

  4. How do I operate with polynomial functions? Let’s review…

  5. The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms & put in descending order

  6. The difference f - g To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms. Distribute negative

  7. The product f • g To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function. FOIL Good idea to put in descending order

  8. The quotient f /g To find the quotient of two functions, put the first one over the second. Nothing more you could do here. (If you can reduce these you should. More later…)

  9. Operations with Polynomials Now you try the four operations on the bottom of page 1 of your handout.

  10. Factoring Review: #1: GCF Method

  11. GCF Method is just distributing backwards!!

  12. Review: What is the GCF of 25a2 and 15a? 5a Let’s go one step further… 1) FACTOR 25a2 + 15a. Find the GCF and divide each term 25a2 + 15a = 5a( ___ + ___ ) Check your answer by distributing. 5a 3

  13. Find the GCF 6x2 Divide each term by the GCF 18x2 - 12x3 = 6x2( ___ - ___ ) Check your answer by distributing. 2) Factor 18x2 - 12x3. 3 2x

  14. 3) Factor 28a2b + 56abc2. GCF = 28ab Divide each term by the GCF 28a2b + 56abc2 = 28ab ( ___+ ___) Check your answer by distributing. 28ab(a + 2c2) a 2c2

  15. 4) Factor 20x2 - 24xy • x(20 – 24y) • 2x(10x – 12y) • 4(5x2 – 6xy) • 4x(5x – 6y)

  16. 5) Factor 28a2 + 21b - 35b2c2 GCF = 7 Divide each term by the GCF 28a2 + 21b - 35b2c2 = 7 ( ___ + ___ - ____ ) Check your answer by distributing. 7(4a2 + 3b – 5b2c2) 4a2 3b 5b2c2

  17. Factor 16xy2 - 24y2z + 40y2 • 2y2(8x – 12z + 20) • 4y2(4x – 6z + 10) • 8y2(2x - 3z + 5) • 8xy2z(2 – 3 + 5)

  18. Factor out the GCF for each polynomial:Factor out means you need the GCF times the remaining parts. a) 2x + 4y 5a – 5b 18x – 6y 2m + 6mn 5x2y – 10xy Greatest Common Factorsaka GCF’s 2(x + 2y) How can you check? 5(a – b) 6(3x – y) 2m(1 + 3n) 5xy(x - 2)

  19. Ex 1 • 15x2 – 5x • GCF = 5x • 5x(3x - 1)

  20. Ex 2 • 8x2 – x • GCF = x • x(8x - 1)

  21. Ex 3 • 8x2y4+ 2x3y5 - 12x4y3 • GCX = 2x2y3 • 2x2y3(4y + xy2 – 6x2)

  22. #2: X-box Factoringaka Diamond Method

  23. X- Box Product 3 -9 Sum

  24. X-box Factoring • This is a guaranteed method for factoring quadratic equations—no guessing necessary! • We will review how to factor quadratic equations using the x-box method • Background knowledge needed: • Basic x-solve problems • General form of a quadratic equation

  25. Standard 11.0 Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. Objective: I can use the x-box method to factor non-prime trinomials.

  26. Factor the x-box way Example: Factor 3x2 -13x -10 (3)(-10)= -30 x -5 3x 3x2 -15x 2 -15 -13 -10 2x +2 3x2 -13x -10 = (x-5)(3x+2)

  27. Factor the x-box way y = ax2 + bx + c Base 1 Base 2 Product ac=mn First and Last Coefficients 1st Term Factor n GCF n m Middle Last term Factor m b=m+n Sum Height

  28. -12 4 Examples Factor using the x-box method. 1. x2 + 4x – 12 a) b) x +6 x26x -2x -12 x 6 -2 -2 Solution: x2 + 4x – 12 = (x + 6)(x - 2)

  29. Examples continued 2. x2 - 9x + 20 a) b) x -4 -4-5 20 -9 x2 -4x -5x 20 x -5 Solution: x2 - 9x + 20 =(x - 4)(x - 5)

  30. Examples continued 3. 2x2 - 5x - 7 a) b) 2x -7 -72 -14 -5 x 2x2 -7x 2x -7 +1 Solution: 2x2 - 5x – 7 = (2x - 7)(x + 1)

  31. Examples continued 3. 15x2 + 7x - 2 a) b) 3x +2 -30 7 5x 15x2 10x -3x -2 10-3 -1 Solution: 15x2 + 7x – 2 = (3x + 2)(5x - 1)

  32. #3: Difference of Squares • a2 – b2 = (a + b)(a - b)

  33. What is a Perfect Square • Any term you can take the square root evenly (No decimal) • 25 • 36 • 1 • x2 • y4

  34. Difference of Perfect Squares x2 – 4 = the answer will look like this: ( )( ) take the square root of each part: ( x 2)(x 2) Make 1 a plus and 1 a minus: (x + 2)(x - 2 )

  35. FACTORING (x – 8)(x + 8)

  36. Example 1 • 9x2 – 16 • (3x + 4)(3x – 4)

  37. Example 2 • x2 – 16 • (x + 4)(x –4)

  38. Ex 3 • 36x2 – 25 • (6x + 5)(6x– 5)

  39. More than ONE Method • It is very possible to use more than one factoring method in a problem • Remember: • ALWAYS use GCF first

  40. Example 1 • 2b2x – 50x • GCF = 2x • 2x(b2 – 25) • 2nd term is the diff of 2 squares • 2x(b + 5)(b - 5)

  41. Classwork Handout pages 3 & 4 odds

  42. Homework M3U3D1 Handout pages 3&4 evens. Show all your work to receive credit– don’t forget to check by multiplying!

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