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Warm - up 6.2 Factor: 1. 4x 2 – 24x4x(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 = 0X = -5

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

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

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

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

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

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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

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

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The polynomial in the last example has three “zeros,” but it only has 2 distinct zeros: -2, 3. 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 repeated zero is called a multiple zero. A multiple zero has a multiplicity equal to the number of times the zero occurs. 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

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

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

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