Part 2: Polynomial Graphing The following problems concern the polynomial p(x) = 23 + 312 - 12x + 7. 9. Complete the following three ways to tell that r = r is a zero of p. (a) The point r = r is an I-. of the graph of y = p(I). (b) The equation p(r) = - is true. (c) is a factor of p. 10. Use method (b) from #9 to verify that r = 1 is a zero of p. 11. Since x = 1 is a zero of p, what must be a factor of p? 12. Divide p(x) by the factor from #11, using polynomial division as in Model 1. Identify the quotient and remainder. 13. Your remainder in #12 should be zero. (Why?) Revise your work if necessary. 14. Now you have partially factored p(r). Fill in the blank below with the quotient from #12. p(x) = 2x3 + 3x2 - 12x + 7 = (x -1)(15. Factor the quotient to write p(x) in its fully factored form. p(x) = 2x3 +3x2 - 12r + 7 = (x - 1)( 16. List each zero of p with its multiplicity. 17. Describe the end behavior of the graph of y = P(). to the right. The graph of y = p(x) to the left and rises / falls rises / falls Explain how you got your answers. 18. Find the y-intercept of the graph of y = P(x). 19. Draw the graph of y = p(x) below. Label the y-intercept and all x-intercepts. Make sure your graph's behavior at each x-intercept matches the multiplicity of that zero. 8+ 6 - 4 2 . -5 -4 - 3 -2 -1 co - 5 -2- -4- -6 - -8 -10 -Model 3: Zeros, Factors, and Multiplicity f(x) = (2-1)(x -2)(2+1)4 g(x) = 2(2+2)(2-3)2 12. Solve the equations f(x) = 0 and g(x) = 0 to find the x-intercepts of the graphs in Model 3. Label the intercepts on the graphs. Note: These x-values are known as zeros of the polynomials. 13. Suppose p is a polynomial with zeros x = -8, x = -5, and x = 6. Write a possible formula for p(2). 14. Complete this sentence: If r = r is a zero of a polynomial p, then (- -) is a factor of p. Definition: If p is a polynomial with zero x = r, then the multiplicity of r is the highest power of (x - r) that is a factor of p. 15. Give the multiplicity of each zero you found in #12. 16. Based on the graphs in Model 3, fill in the blanks below. (a) If the multiplicity of a zero of a polynomial is even / odd then the polynomial's graph crosses the x-axis at that point. (b) If the multiplicity of a zero of a polynomial is even / odd then the polynomial's graph touches but does not cross the r-axis at that point.Definition: A polynomial function has the form f(x) = and" + an-12" - + .. . + azz' + aji + do where n is a non-negative integer and the coefficients an, . .., do are real numbers. 3. Check your answers to #2 to see if they agree with the above definition. Revise them if necessary. 4. Use the examples in the table below to define the following terms. Leading Polynomial Degree Leading Term Constant Term Coefficient 213 - 3x - 2 2x-3 1x2 - 2x4 + 27 - 5x -274 -2 -75 L 0.126 - 2x - 12 0.176 01 -12 (a) The degree of a polynomial is (b) The leading term of a polynomial is (c) The leading coefficient of a polynomial is (d) The constant term of a polynomial is Reporter: Check your answers to #4 with a nearby team before moving on to #5. 5. The graphs of three polynomial functions are given below. Identify the degree of each polynomial. (a) (b) (c)6 Polynomials Model 1: Polynomials and Non-polynomials These are examples of polynomials: These examples are not polynomials: . 5x2 + 3x - 7 . 3x + 5VI - 2 . 25.9x2 - 0.34x7 - 14.213 . 2375 +8 . 24 + V2x+ 1 - 20 2 23 + 22 1. Based on the examples in Model 1, identify which of the following are polynomials. Hint: Four of them are polynomials. (a) Vx2 +1 (f) 25 -24+3-12+ (b) 1 + vor - x (8) 24+ 2-3 (c) 8.3x - TI2 (h) 2:5/3 - 15 (d) 12 (i) x +0.57 (e ) 3 7 2 (j) 5x - 22 + 2 3 Reporter: Check your answer to #1 with a nearby team before moving on to #2. 2. Based on your work so far, finish the following sentences. Be as specific as possible. (a) In a polynomial, the powers of x can be (b) In a polynomial, the coefficients can be (c) The domain of a polynomial function is3 if x 1 each segment. if a