5.2 Polynomials and Linear Factors (Day 1)

Prime Factors vs. Linear Factors 12 Standard Form Factored Form (Linear Factors) 4 3 2 2 Linear Factors are similar to Prime Factors in that they cannot be ________ any further. factored

Writing a Polynomial in Standard Form Ex 1.

Writing a Polynomial in Factored Form Ex 2. Ex 3. GCF!!!

3x(x2 – 6x + 8) 3x(x – 4)(x – 2) x(x2 – 7x – 18) x(x + 2)(x – 9) Write each polynomial in factored form. Check by multiplication. White Boards 4. 3x3 – 18x2 + 24x 5. x3 – 7x2 – 18x 3x(x2 – 6x + 8) 3x(x – 4)(x – 2) x(x2 – 7x – 18) x(x + 2)(x – 9)

Finding Zeros of a Polynomial Function 6. Find the zeros of y = (x + 1)(x – 1)(x + 3). Then the graph of the function. (Include all min/max points) Use the zero-product property. x = -1, 1, -3 Max Min Graph! (-2.2, 3.1) (0.2, -3.1)

7. y = 2x(x + 2)(x – 3) 8. y = –(x + 4)(x + 1) Find the zeros of each function. Then graph the function. (Make sure you include the min/max points.) White Boards 7. y = 2x(x + 2)(x – 3) 8. y = –(x + 4)(x + 1) Find the relative maximum,relative minimum, and the zeros of each function. 9. f(x) = –2x3 –12x2+40x–12 Together 1st

Theorem Zeros = a, b, - c Polynomial Function y = (x – a) (x – b) The expression x – a is a linear of a polynomial if and only if the value of a is a zero of the related polynomial function. Factor Theorem: Zeros = a, b, - c Polynomial Function y = (x – a) (x – b) (x + c)

Writing a Polynomial Function From its Zeros 1. Write a polynomial function in standard form with zeros at 2, -3, and 0. 2 -3 0 f(x) = (x – 2)(x + 3)x y = (x2 + 3x – 2x – 6)x y = (x2 + x – 6)x y = x3 + x2 – 6x

Write a polynomial function in standard form with the given zeros. White Boards 2. x = 0, 0, 5 3. x = 2, -2, 4

Terms Multiple Zero: Multiplicity: number of times the multiple zero occurs a repeated zero

Finding the Multiplicity of a Zero 4. Find any multiple zeros of f(x) = x5 – 6x4 + 9x3 and state the multiplicity. f(x) = x5 – 6x4 + 9x3 = x3(x2 – 6x + 9) = x3(x – 3)(x – 3) 0 with multiplicity 3 with multiplicity Zeros: 3 odd multiplicity: pass through the x-axis 2 even multiplicity: does NOT pass through the x-axis, it doubles and goes back in the direction it came from Look at the graph. (Explain the difference between odd and even multiplicity.)

5. y = 9x5 – 36x3 6. y = (x + 2)4(x – 3) White Boards Find the zeros of each function. State the multiplicity of multiple zeros. White Boards 5. y = 9x5 – 36x3 6. y = (x + 2)4(x – 3)

Terms is the biggest y-value among the nearby points on the graph is the least y-value among the nearby points on a graph Relative Maximum: Relative Minimum:

What is this a picture of? What part of the graph is important? Discuss minimum, maximum, x-intercepts (zeros)

Summary 1. -4 is a solution of x2 + 3x – 4 = 0 2. -4 is an x-intercept of the graph of y = x2 + 3x - 4 3. -4 is a zero of y = x2 + 3x – 4 (also called a root) 4. x+4 is a factor of x2 + 3x - 4