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Published byLuciano Corin Modified over 2 years ago

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

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

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

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

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U – SUBSTITUTION: au 2 + bu + c = 0 Step #1: Must have a trinomial in which one power of x is DOUBLE the other. ax 2n + bx n + c = 0 Step #2: Let u equal smaller exponent of x u = x n Step #3: SUBSTITUTE u into the trinomial to create a quadratic equation. Step #4: Use FACTORING or QUADRATIC FORMULA to find roots for u and solve for x n. u = Root #1 and u = Root #2 x n = Root #1 and Root #2

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

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c) 7x 10 – 6 = 0 d) x 7 + 2x = 0 Not Possible: x 7 is more than double x 2 power 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) 2x 6 + x = 0 b) x 4 + 2x = 0 Let u = x 2, u 2 + 2u + 10 = 0 Let u = x 3, 2u 2 + u + 9 = 0 Not Possible: No second x term to use Let, u 2 - 2u + 8 = 0 Let, u 2 - 5u + 10 = 0

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

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

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

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

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

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