Warm - up 6.2 Factor: 1. 4x 2 – 24x4x(x – 6) 2. 2x x – 21(2x – 3)(x + 7) 3. 4x 2 – 36x + 81 (2x – 9) 2 Solve: 4. x x + 25 = 0X = -5

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

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

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

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

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.

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

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

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

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

6.2 Homework Page (1-11 odd, odd, all)