TEXTBOOK HELP Pg. 313 Chapter 3.2-3.4 REVIEW ANSWER KEY 1. What qualifies a function as a polynomial? Powers = non-negative integers Polynomial functions of degree 2 or higher have graphs that are smooth and continuous. Give 2 examples of a polynomial: Give 3 examples of NOT a polynomial: x 5 +3x + 2 3x 3 2x 1/2 8/x 2x 2 Pg. 314-315 2. How do you determine end behavior? Find the highest power in the polynomial. That determines the end behavior. If an odd exponent, same shape as x 3. If even exponent, same shape as x 2. Negative leading coefficients will make it flip upside down. Another way to describe: Leading coefficient determines the right side. If positive, it goes up. Negative goes down. If the exponent is even, then the left side will do the same thing. If the exponent is odd, then the left side does the opposite. Determine the end behavior of the following polynomial functions? f(x) = -3x 4 + 5x 8 + 6x 7 g(x) = -4x 7 + 8x -10 h(x) = 5x 9 (x+2) 2 (x-3) 2 Leading coeff: 5x 8-4x 7 5x 13 x +, f(x) + x +, f(x) x +, f(x) x -, f(x) x -, f(x) x -, f(x) Pg. 331-332 3. Divide using Synthetic Division. Write the answer in standard form including the remainder. Then, state what point would be included on the graph. = 6x 2-8x +11- -1 6-2 3 4-6 8-11 6-8 11-7 f(x) = (x+1)(6x 2-8x +11- (-1, -7) 4. = -2x 3-6x 2-12x -31-3 -2 0 6 5-10 -6-18 -36-93 -2-6 -12-31 -103 f(x) = (x-3)(-2x 3-6x 2-12x -31 - (3, -103)
Pg. 339 5. How do you determine the possible rational zeros of a function? Is it possible to have zeros that are not rational? If so, what kind of zeros are they? List all the possible rational roots for f(x) = 9x 4 + 3x 3 + 1x + 2 1, 2, -1, -2,- 6. Without a calculator, list all the possible zeros of the function and use synthetic division to find all the zeros: f(x) = x 3-5x 2 + 4x -20 Pg. 340 Ex #4 Possible zeros: 1, 2, 4, 5, 20, -1, -2, -4, -5, -20 2 1-5 4-20 2-6 -4 1-3 -2-24 NOT A ZERO (student may try other problems that don t work until they find a zero) 5 1-5 4-20 5 0 20 1 0 4 0 This is a zero! So, rewrite f(x) and continue factoring! f(x) = (x-5)(x 2 + 4) 0 = (x-5)(x 2 + 4) zpp 0 = x-5 0 = x 2 + 4 x=5-4 = x 2 ZEROS: (5, 0) (2i, 0) (-2i, 0) Is there a another way to find the zeros of #6. If so, what is it and which way do you think is faster? YES. This problem can be factored using grouping. 7. Given the function f(x) = 4x (x + 4) 5 (x + 2)(x - 1) 4, state the zeros and their multiplicity. Then explain whether each point will touch or cross the graph. Zeros Multiplicity Touch or Cross? Pg. 319 0-4 -2 1 1 5 1 4 Cross Cross Cross Bounce
8. Use your calculator to sketch a quick graph of each function and label the domain and range. (HINT: may need to use the MIN and MAX functions) F(x) = 3x 5 6x g(x) = 2x 4-8x 2 Domain: Range: Domain: Range:
Pg. 317 Ex #4 GRAPH THE FUNCTION. Make sure to show any and all calculations and/or give a brief explanation of how you came to your answer. 9. f(x) = x 3 4x 2 + 8x - 32 ANSWERS HERE End Behavior: x +, f(x) x -, f(x) WORK / EXPLAIN: 1x 3 Leading coefficient = +1 right side up Odd degree 3 Left side opposite All Possible Rational Zeros: +1, 2, 4, 8, 16, 32, -1, -2, -4, -8, -16, -32 Y-Intercept: (0, -32) f(0) = 0 3 4(0) 2 + 8(0) 32 = -32 X-Intercepts Multiplicity Cross/Bounce ( ) 1 cross 1 cross (4, 0) 1 cross f(x) = x 3 4x 2 + 8x 32 0 = (x 3 4x 2 )(+ 8x 32) 0 = x 2 (x 4 ) +8(x 4) 0 = (x 2 +8)(x-4) zpp 0 = x 2 +8 0=x-4 GRAPH: (make sure to label the key points you just looked found) x=4 x=4 x=4
Pg. 341 Ex #5 f(x) = x 3 + 11x 2 + 40x + 48 ANSWERS HERE End Behavior: x +, f(x) x -, f(x) All Possible Rational Zeros: WORK / EXPLAIN: 1x 3 Leading coefficient = +1 right side up Odd degree 3 Left side opposite Y-Intercept: (0, 48) f(0) = 0 3 + 11(0) 2 + 40(0) + 48 = 48 X-Intercepts Multiplicity Cross/Bounce First check to see if you can factor the polynomial. -3 1 cross -4 2 bounce 0 = (x 3 + 11x 2 ) (+ 40x + 48) x 2 (x + 11) + 8 (5x + 6) Since this cannot be factored, we ll have to use synthetic division. Use your calculator to find the actual rational zeros. X=3 (note: I set the window to for x is -6 to 5 and -2 to 2 for y) Then, use synthetic division to finish factoring. GRAPH: (make sure to label the key points you just looked found) -3 1 11 40 48-3 -24-48 1 8 16 0 f(x) = (x+3)(x 2 + 8x +16) f(x) = (x+3)(x+4)(x+4) 0 = (x+3)(x+4)(x+4) 0 = x+3 0=x+4 (double root) x= -3 x= -4
8. Describe how to determine the function given the zeros:each zero relates to a factor and can fit into the equation f(x) = a(x-p)(x-q)..then, plug in another point to determine the a value. Finally, write the equation. What happens when you are given one conjugate? Every imaginary number has a conjugate. So, if the given information lists that i is a zero, there also exists a point at i. Same thing for something like: 5+2i. Its conjugate, 5+2i, is also a zero. Pg. 344 Ex #6 a. zeros: 1, 5 (double root), -6; y intercept is -75 f(x) = a (x-1)(x-5) 2 (x+6) b. zeros: -2, 3, and i f(5) = 728 f(x) = a (x+2)(x-3)(x-i)(x+i) -75 = a (0-1)(0-5) 2 (0+6) f(x) = a (x+2)(x-3)(x 2 +1) -75 = a (-1)(25)(6) 728 = a (5+2)(5-3)(5 2 +1) -75 = a (-150) 728 = a (7)(2)(26) -1/2 = a 728 = a 364 2 = a c. zeros: 3, 5+I f(x) = a(x-3)(x-(5+i)) (x-(5-i)) f(x) = a(x-3)(x-5-i)) (x-5+i)) f(x) = a(x -3)(x 2 10x +26) ***This problem needed to have given us another point in order to finish. Since it did not, this is as far as we can go. 1. Use synthetic division to evaluate f(2) if f(x) = 3x 4 2x 2 x 100. 2 3 0-2 -1-100 6 12 20 38 3 6 10 19-62 f(2) = -62 You can (and should) check your answer by f(2) = 3(2) 4 2(2) 2 2-100 = -62, but make sure to show synthetic division if it is in the instructions.
2. List the possible zeros of 3x 3 2x 2 + 9x 6 3. Given that f(x) = x 4 x 3 8x 2 4x 28 and plugging the function in the calculator reveals that x=-3 and x=4 are zeros of the function, find all the other zeros. ****THIS PROBLEM IS WRITTEN INCORRECTLY. PLEASE DISREGARD, however, be ready to answer one like this on the test..