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CST Prep Part II MAIN MENU

Solve polynomial problems involving area expressions and factoring. Learn vocabulary, rules, and strategies. Includes solved examples and explanations. Designed by Ms. Carranza and Mrs. Murray.

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CST Prep Part II MAIN MENU

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  1. CST Prep Part IIMAIN MENU Designed by Ms.Carranza and Mrs. Murray Solved by: 8th Grade Gate Students 2011

  2. Standard 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques. Problem 47 Problem 48 Problem 49 Main Menu Problem 51 Problem 50 Problem 52

  3. Vocabulary Rules & Strategies 51) A volleyball court is shaped like a rectangle. It has a width of x meters and length of 2x meters. Which expression gives the area of the court in the square meters? A 3x B 2x² C 3x² D 2x³ Solution & Answer Standard 10

  4. Vocabulary • Expression : a mathematical phrase that contains operations, numbers, and/or variables. • Area: The number of non overlapping unit squares of a given size that will exactly cover the interior of a plane figure. • Square Meters: a unit of area measurement equal to a square measuring one meter on each side. Back to Problem

  5. Rules & Strategies • Area= base (length) * height (width) • Product of Powers: combine base, add powers • Add exponents ( x = x ) 1 x 2x Back to Problem

  6. Solution 2x • 1x 2x Answer: B 1 1 2 Back to Problem Standard 10

  7. Vocabulary Rules and Strategies 50) Which of the following expressions is equal to (x+2)+(x-2)(2x+1)? 2x² - 2x 2x² - 4x 2x² + x 4x² + 2x Standard 10 Solution and Answer

  8. Solution and Answers(x+2)+(x-2x)(2x+1)? Step 1: Area Method Step 2: Combine like terms and put in Descending order Answer: 2x² - 2x 2x + 1 x -2 2x²-3x-2 1X + 2 + 2x² - 3x - 2 2x² +x -4x -2 0+ 2x² -2x Standard 10 Back to Problem

  9. Vocabulary Area Method: ax + b (ax + b)(ax + b) ax b aax + (abx + abx) + bb Descending Order: ordering terms from greatest to least (ex. 1x+ 2x² + 3x+1x³ + 9 = 1x³ + 2x² + 3x + 1x + 9) aax abx abx bb Back to Problem

  10. Rules & Strategies 1st.Use the Area Method 2nd.Compare like terms 3rd.Descending Order Back to Problem

  11. Vocabulary Rules & Strategies 49)The sum of the two binomials is 5x2-6x, If one of the binomials is 3x2-2x, what is the other binomial A 2x2-4x B 2x2-8x C 8x2+4x D 8x2-8x Solution and Answer Standard 10

  12. Vocabulary • sum– answer t an addition problem • Binomial- A polynomial with two terms Back to Problem

  13. Rules and strategies • Line up the binomials • Combine both binomials by subtracting Back to Problem

  14. Solution (3x2-2x) +2x2-4x 5x2-6x Answer: A 1st binomial 2nd binomial Sum of binomials Back to Problem Standard 10

  15. Vocabulary Rules & Strategies 47) 5x310x7 A) 2x4 B)1 2x4 C) 1 5x4 X4 5 Solution and Answer Standard 10.0

  16. Vocabulary • Quotient of Powers: simplify coefficients combine base subtract exponents Back to Problem

  17. Rules & Strategies • In order to skip the negative exponent rule, circle the biggest exponent to tell if the variable stays in the denominator or numerator. • Subtract powers • Divide if there is any whole numbers in the numerator and denominator. Back to Problem

  18. Solution 5x3 1 10x7 2x4 Answer: B Back to Problem Standard10.0

  19. Vocabulary Rules and Strategies 48.) (4x²-2x+8) – (x²+3x-2) A) 3x²+x+6 B) 3x²+x+10 C) 3x²-5x+6 D) 3x²-5x+10 Solution and Answer Standard 10

  20. Vocabulary • Parenthesis ( ); indicates separate grouping of symbols • Exponent ²; a symbol or number placed above and after another symbol or number to denote the power to which the latter is to be raised • Variable ‘x’; a quantity or function that may assume any given value or set of values Back to Problem

  21. Rules and Strategies • Subtracting Polynomials / Lesson 7-7 • Distribute (-) before grouping Back to Problem

  22. Solution and Answer (4x²-2x+8) – 1 (x²+3x-2) (4x²-2x+8)-x²-3x+2 (4x²-1x²) + (-2x-3x) + (8+2) 3x²-5x+10 Answer:D Back to Problem Standard 10

  23. Standard 11 Students solve multistep problems involving linear equations and inequalities in one variable. Problem 53 Problem 54 Main Menu Problem 55 Problem 56

  24. Vocabulary Rules & Strategies 53)Which is the factored form of 3a²-24ab+48b²? a. (3a-8b)(a-6b) b. (3a-16b)(a-3b) c. 3(a-4b)(a-4b) d. 3(a-8b)(a-8b) Solution & Answer Standard 11

  25. Vocabulary • Factored form= Form of equation in which each term is simplified from factoring methods and GCF. Back to Problem

  26. Rules & Strategies • Ask yourself if you can factor out a GCF • Use the diamond method • Keep asking yourself whether you can factor more. If there is a GCF, don’t forget to Include it in your final answer. Back to Problem

  27. Solution & Answer Factor out a gcf 3a²-24ab + 48b² 3 3 3 3(a²-8ab+16b²) +16 keep factoring!! a a -4b -4b 3 (a-4b)2 Solution: C -8 Standard 11 Back to Problem

  28. Vocabulary Rules & Strategies 54) Which is the factor of x² – 11x + 24 A x + 3 B x - 3 C x + 4 D x - 4 Solution & Answer Standard 11

  29. Vocabulary • Factor = A number or expression that is multiplied by another number or expression to get a product. • Term = a part of an expression that is added or subtracted • Binomial = 2 terms • Trinomial = 3 terms Back to Problem

  30. Rules & Strategies • Since there are three terms in this trinomial you use the “Diamond Method” • Diamond Method Formula : ax² + bx + c • Make an “X” on your paper. • On the top intersect write a multiplication sign (x) to show that the two denominators multiply to equal the number you're going to get above the multiplication sign. • On the bottom intersect write a plus sign (+) to show that the two denominators added together equal the number you’re going to get below the addition sign. • Plug the value for “c” above the multiplication sign ( +24 ) • Plug in the value for “b” below the addition sign ( -11 ) • Now , think of two numbers that multiplied equal +24 and added equal -11 . • ( You should come up with -3 and -8 ) • Lastly , you rewrite your fractions as binomials : 1x = ( x – 3 ) -3 Back to Problem

  31. Solution x² – 11x + 24 + 24 1x x 1x -3 + -8 - 11 ( x – 3 ) ( x – 8 )  Final answer Answer: B Back to Problem Standard 11

  32. Vocabulary Rules & Strategies 55) Which of the following shows 9t + 12t + 4 factored completely? 2 2 A (3t + 2) B (3t + 4) (3t +1) C (9t + 4) (t + 1) D 9t + 12t + 4 2 Solution & Answer Standard 11

  33. Vocabulary • Factored :one of two or more numbers, algebraic expressions, or the like, that when multiplied together produce a given product; a divisor. Back to Problem

  34. Rules & Strategies • Always ask yourself which factoring method should I use? • You should always see if you can factor out a GCF • Use“diamond method” when there is no GCF and the coefficient is greater than one. Back to Problem

  35. Solution • Make sure to simplify the fractions 2 9t + 12t + 4 +36 3 9t 9t 3 3t 3 +6 +6 3 2 +12 (3t+2) 3t 2 Answer: A Back to Problem Standard 11 2

  36. Vocabulary Rules & Strategies 56) What is the complete factorization of 32-8z² -8(2+z)(2-z) 8(2+z)(2-z) -8(2+z)² 8(2-z)² Solution & Answer Standard 11

  37. Vocabulary • Factorization: to simplify Back to Problem

  38. Rules & Strategies • Find GCF • Divide each term by GCF • Rewrite equation w/ GCF outside of parenthesis • Divide inside terms by a negative • Rewrite the equation w/ negative on the outside • Difference of 2 Squares • Rewrite new equation & THAT’S YOUR ANSWER ! Back to Problem

  39. Solution 32-8z² -8 -8 factor out a -8 to get a positive z² -8(z²-4) keep factoring diff of 2 squares -8(z+2)(z-2) Or -8(2+z)(2-z) Answer: A. -8(2+z)(2-z) Standard 11 Back to Problem

  40. Standard 12 • Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. Problem 77 Problem 78 Problem 79 Problem 80 Problem 81 Main Menu

  41. Vocabulary Rules and Strategies 78.) 6x2 + 21x + 9 4x2 - 1 A)3(x+1) 2x-1 B) 3(x+3) 2x-1 C) 3(2x+3) 4(x-1) D) 3(x+3) 2x+1 Solution and Answer Standard 12

  42. Vocabulary • GCF ; Greatest Common Factor • Super Diamond ; x2 + bx + x • Difference of Two Squares ; terms need to be in perfect squares [Example: (a+b)(a-b)] Back to Problem

  43. Rules and Strategies • Top ; find GCF, super diamond • Bottom ; difference of two squares, terms need to be perfect squares Back to Problem

  44. Solution and Answer 6x2 + 21x + 9 4x2 – 1 1) GCF 2) Difference of Two Squares 3)Cross Cancel Top/Numerator ; 6x2 + 21x + 9 Top/Numerator ; 6x2: 2 ∙ 3 ∙ x ∙ x 21x: 3 ∙ 7 ∙ x 9: 3 ∙ 3 GCF: 3 6x 2 +21 + 9 3 3 3 3 ( 2x 2 + 7x + 3) Super Diamond!! Bottom/Denominator ; 4x2 - 1 +6 2x 2x 1 1 x +2x +2x = ∙ + (2x+1)(2x-1) +6 +1 +3 Cross Cancel 3(x+3)(2x+1) (2x+1)(2x-1) +7 Answer ; B 3(x+3) 2x-1 = 3(x+3)(2x+1) Back to Problem Standard 12

  45. Vocabulary Rules & Strategies 79)What is x2-4x+4 reduced to lowest terms? x2-3x+2 x-2 X-1 B) x-2 X+1 C) x+2 X-1 D) x+2 X+1 Solution & Answer Standard 12

  46. Vocabulary • Reduce = lower in degree • Lowest term = the form of a fraction after dividing the numerator and denominator by their greatest common divisor. Back to Problem

  47. Rules & Strategies Back to Problem • First you should ask yourself which strategy is best to change the equation into a dividable state using either; Greatest common facture (GCF), difference of 2 squares, or one of the diamond methods. • In this case the best one would be the diamond method on both the denominator and numerator. • Once you find the factors of both of the trinomials reduce by dividing the like terms.

  48. Solution ax2+bx+c=0 Step 1: x2-4x+4 x2-3x+2 Step 2: x2-4x+4 (x-2) (x-2) x2-3x+2 (x-2) (x-1) Diamond Method 4 2 x * xx * x -2 -2 -2 -1 + + -4 Divide out! -3 (x-2) (x-2) = (x-2) Answer: A (x-2) (x-1) (x-1) Back to Problem Standard 12

  49. Vocabulary Rules and Strategies 80. What is 12a3 – 20a2 reduced to lowest terms? 16a2 + 8a A) a 2 B) 3a – 5 2a + 1 C) -2a 4 + 2a D) a (3a – 5) 2 (2a + 1) Solution and Answer Standard 12

  50. Vocabulary • Reduced ; simplified or lowest terms Back to Problem

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