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Warm-Up

DEGREE is Even/Odd, LEADING COEFFICIENT is Positive/Negative, END BEHAVIOR EXTREMA (Max or Min, Relative or Absolute). Warm-Up. (-2, 15). EVEN (2) POS (3) END BEHAVIOR x  ±∞; f(x)  ∞ (4) A. Min: (7, -21) R. Min: (-9, -8) R. Max: (-2, 15). ODD (2) NEG

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Warm-Up

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  1. DEGREE is Even/Odd, • LEADING COEFFICIENT is Positive/Negative, • END BEHAVIOR • EXTREMA (Max or Min, Relative or Absolute) Warm-Up (-2, 15) • EVEN • (2) POS • (3) END BEHAVIOR x  ±∞; f(x)  ∞ • (4) A. Min: (7, -21) • R. Min: (-9, -8) R. Max: (-2, 15) • ODD • (2) NEG • (3) END BEHAVIOR x  - ∞; f(x)  ∞ • x  ∞; f(x) - ∞ • (4) R. Min: (-3, 4) R. Max: (8, 9) [1] [2] (8, 9) (-3, 4) (-9, -8) (7, -21) [3] [4] • EVEN • NEGATIVE • END BEHAVIOR • Absolute Max:Relative Max:Relative Min: • ODD • POSITIVE • END BEHAVIOR • Relative Max:Relative Min:

  2. Factoring Polynomials Review: [1]Difference of SQUARES Example [2]Difference of CUBES Example [3]Sum of CUBES Example

  3. [4]Factoring Trinomials: ax2 + bx + c Example Multiply = -12| Add = -4 -6 * 2 = 12; -6 + 2 = -4 Step #1:Find the factor pair (n1and n2) that MULTIPLY = ac(outsides) and ADD = b(middle). Step #2:Split the middle termbx = n1x + n2x Step #3:Perform factor by grouping on ax2 + n1x + n2x + c GCF of ax2 + n1x and GCF n2x + c = (?x + ?) (?x + ?)

  4. Factoring Polynomials: PRACTICE a) b) c) e) d) f)

  5. U – SUBSTITUTION: Step #1:Must have a trinomial in which one power of x is DOUBLE the other. ax2n + bxn + c = 0 Step #2:Let u equalsmaller exponent of x u = xn Step #3:SUBSTITUTE u into the trinomial to create a quadratic equation. au2 + bu + c = 0 Step #4:Use FACTORING or QUADRATIC FORMULA to find roots for u and solve for xn. u = Root #1 and u = Root #2  xn = Root #1 and Root #2

  6. EXAMPLE of U – SUBSTITUTION: • x4 – 16x2 + 60 = 0 • Step #1: x4 is double the x2 exponent • Step #2: u = x2 • Step #3: u2 – 16u +60 = 0 • Step #4: Solve u2 – 16u +60 = 0 • Factoring: (u – 10)(u – 6)=0 • Roots: u = 10 and u = 6  x2=10 and x2=6 • Solve for x:

  7. e) f) Example 1: Quadratic Form Only If possible, identify the variable term for u and write each equation in quadratic form using U-SUBSTITUTION. a) 2x6 + x3 + 9 = 0 b) x4 + 2x2 + 10 = 0 Let u = x3, 2u2 + u + 9 = 0 Let u = x2, u2 + 2u + 10 = 0 d)x7 + 2x2 + 10 = 0 c)7x10 – 6 = 0 Not Possible: No second x term to use Not Possible: x7 is more than double x2 power Let , u2 - 2u + 8 = 0 Let , u2 - 5u + 10 = 0

  8. b) Example 2: Solve using U-SUBSTITUTION Check to factor substituted quadratic form. a)

  9. Example 2: U-Substitution Part 2 Check to factor substituted quadratic form. c) d)

  10. f) Example 2: U-Substitution Part 3 Check to factor substituted quadratic form. e) Cannot square -4 …because there is no number that multiples by itself to equal -4

  11. Example 2: U-Substitution Part 4 Check to factor substituted quadratic form. g) h) i) j)

  12. Example 3: Solving Equations of Perfect Cubes Factor and Apply Quadratic Formula b) a) c) d)

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