ALGEBRA II HONORS/GIFTED @ ALGEBRA II HONORS/GIFTED - SECTION 5-2 (Polynomials, Linear Functions, and Zeros) 2/17/2019 ALGEBRA II HONORS/GIFTED @ SECTION 5-2 : POLYNOMIALS, LINEAR FUNCTIONS, and ZEROS

With your graphing calculator, graph f(x) = x3 – 2x2 – 15x. 1) What are the x-intercepts? ANSWERS : -3, 0, 5 2) Now, substitute -3, 0, and 5 into f(x). What happens? ANSWER : f(x) = 0 4) Finally, solve using the factors you found in number 3. 3) Factor f(x). ANSWER : x(x – 5)(x + 3) ANSWERS : 0, 5, -3

The summary………… Given the polynomial P(x), if x – b is a linear factor then *b is a zero of P(x) *b is a root of P(x) *b is an x-intercept of P(x). FACTOR THEOREM : x – b is a factor of P(x) if and only if b is a zero of P(x).

RELATIVE MINIMUM : The value of a function at a down to up point. Let’s go back and find the relative minimum and relative maximum of P(x) using the MAXIMUM and MINIMUM functions of your calculator. RELATIVE MAXIMUM : The value of a function at an up to down turning point. RELATIVE MINIMUM : The value of a function at a down to up point. See page 291 of your book.

6) Find the zeros of y = (x – 3)(x + 4)(x – 3) 5) Factor. x3 + x2 – 12x ANSWER : x(x + 4)(x – 3) 6) Find the zeros of y = (x – 3)(x + 4)(x – 3) ANSWER : 3, -4 We can say that 3 has a multiplicity of 2. That is, 3 appears twice as a zero. Negative four has a multiplicity of one.

7) What is a cubic polynomial function in standard form with zeros 1, -1, and 4? ANSWER : x3 – 4x2 – x + 4 8) What are the zeros of f(x) = x3 – 5x2 + 3x + 9? What are their multiplicities? How do the graphs behave at the zeros? ANSWERS : 3 with a multiplicity of 2, there is a vertex at that point; -1 with a multiplicity of one, the graph crosses through the x-axis at that point.

Why did the imaginary number turn red? It ran out of i-drops.