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Lesson 1.2 Polynomial Functions Workbook 1.2 Explore Your Understanding Assignment 1.2 This assignment includes 18 marks. You are expected to attempt all questions showing detailed solutions. If you are not confident with some of your work, then you may choose to designate up to 3 marks worth to be exempted from grading by writing "DO NOT MARK" in the left margin next to the question. You are still expected to show any work you have tried and/or explain your thinking so that we can give you feedback on your response. No questions should be left blank. Please contact your teacher if you have any questions. 1. A corresponding long division and synthetic division are shown. x2 +x - 6 x + 1)x'+ 2x2 - 5x+ 7 1 2 - 5 7 *+ x 2 to 1 1 - 6 x2 - 5x 1 1 - 6 13 - 6x + 7 -6x - 6 13 2 a. Complete the table so the long division and the synthetic division show the same stage of progress. Your solutions will be partially complete divisions. Long Division Synthetic Division x+x - 6 *+ 1 x' + 2x2 - 5x + 7 x2 - 5x - 6x + 7 1 2 -5 7 ADLC Mathematics 30-1 37Workbook 1.2 Lesson 1.2 Polynomial Functions 1 b. Express the division in the form- P(x ) = (x ) + R x - a x - a 1 c. How does the remainder compare to - K ? What does the remainder of this division represent? x - a\forkbook 1.2 Lesson 1.2 Polynomial Functions 3 . This question is designed to generate understanding for why the remainder theorem and factor theorem work. 1 a. Show that- P(x) x - a -= 0(x) + R x - a can be rearranged to P(x) = Q(x)(x - a) + R. 1 b. Evaluate P(a), showing it simplifies to R. 1 c. How does the answer in part b. relate to the remainder theorem? (1) d. Use P(x) = Q(x)(x - a) + R to explain the factor theorem. (Hint: What is the value of R if there is no remainder? Substitute this value for R into O(x)(x - a) + R to show x - a is a factor of P(x).) 40 ADLC Mathematics 30-1Lesson 1.2 Polynomial Functions Workbook 1.2 4. For function f , the graph of y = f(x) extends down into quadrant Ill and down into quadrant IV, the zeros of f are -2, 3, and 5, and the degree of f = 5. 1 a. What degree(s) is/are possible for f? Explain. 1 b. What are the maximum and minimum multiplicities possible for a zero of f? Explain. (1) c. Sketch two possible graphs of y = f(x).d. State two possible equations for f. 5. A cube with side length x has been modified by adjusting its length, width, and height. A student determined a function of the form V(x) = (x + a)(x + b)(x + c) can be used to represent the volume of the new box for different values of x, and then she graphed the function. 10 -9 -8 IN W A M O - 2shift Lesson 1.2 Polynomial Functions Workbook 1.2 a . (1 ) Based on the graph, predict how the side lengths of the original cube changed. (Hint: What do the zeros of the function represent?) 1 ) b. Determine a polynomial function of the form V(x) = (x +a)(x + b)(x +c) that matches the function in the graph. 1 c. Which part(s) of the graph are realistic? Explain. (Hint: All side lengths must be positive. Using two negative side lengths to give a positive volume is not realistic.)