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Warm - up 6.2

Warm - up 6.2. Factor:. 1. 4x 2 – 24x. 4x(x – 6). 2. 2x 2 + 11x – 21. (2x – 3)(x + 7). 3. 4x 2 – 36x + 81. (2x – 9) 2. Solve:. 4. x 2 + 10x + 25 = 0. X = -5. 6.2 Polynomials and Linear Factors. Objective – To Analyze the factored form of a polynomial. CA State Standard

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Warm - up 6.2

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  1. Warm - up 6.2 Factor: 1. 4x2 – 24x 4x(x – 6) 2. 2x2 + 11x – 21 (2x – 3)(x + 7) 3. 4x2 – 36x + 81 (2x – 9)2 Solve: 4. x2 + 10x + 25 = 0 X = -5

  2. 6.2 Polynomials and Linear Factors Objective – To Analyze the factored form of a polynomial CA State Standard - 4.0 Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. - 10.0Students graph quadratic functions and determine the maxima, minima, and zeros of the function.

  3. Write the expression as a polynomial in standard form. Example 1 (x + 1)(x + 1)(x + 2) Multiply last two factors (x + 1)(x2 + 2x + x + 2) Combine 2x and x (x+ 1)(x2 + 3x + 2) Distribute x, then 1 x3 +3x2 + 2x + x2 + 3x + 2 Combine like terms x3 +4x2 + 5x + 2

  4. Write the expression in factored form. Example 2 3x3 – 18x2 + 24x 3x(x2 – 6x + 8) Factor out GCF 3x 3x(x – 2)(x – 4) Factor trinomial using x-box

  5. Relative Max – A point higher than all nearby points. Relative Min – A point lower than all nearby points. Relative Max Relative Min x-intercepts

  6. Finding the zeros of a polynomial function in factored form (use zero product property and set each linear factor equal to zero) Remember: the x-intercepts of a function are where y = 0, these values will now be referred to as the “zeros” of the polynomial. Example 3 y = (x + 1)(x – 3)(x + 2) x + 1 = 0 x – 3 = 0 x + 2 = 0 x = – 1 x = 3 x = – 2

  7. You can write linear factors when you know the zeros. The relationship between the linear factors of a polynomial and the zeros of a polynomial is described by the Factor Theorem. Factor Theorem: The expression x – a is a linear factor of a polynomial if and only if the value “a” is a zero of the related polynomial function

  8. Example 4 Write a polynomial function in standard form with zeros at -2, 3, 3. -2 3 3

  9. The polynomial in the last example has three “zeros,” but it only has 2 distinct zeros: -2, 3. • A repeated zero is called a multiple zero. • A multiple zero has a multiplicity equal to the number of times the zero occurs. (x – 2)(x +1)(x +1) has 3 zeros: 2, -1, -1 since -1 is repeated, it has a multiplicity of 2 Example 5

  10. 6.2 Guided Practice Page 323 – 325 1, 8, 16, 29, and 30

  11. 6.2 Homework Page 323-325 (1-11 odd, 17-35 odd, 65-72 all)

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