Name: Date: Chapter 2 Test Review Pre-Calculus Honors 1. Perform the indicated operations and write the result in standard form. 3+4𝑖 a. (8 − 3𝑖) − (17 − 7𝑖) c. 4−2𝑖 b. (3 − 4𝑖)2 d. 4+√8 2 2. Solve. Express your solution in standard form, if necessary. 𝑥 2 − 2𝑥 + 4 = 0 3. The function 𝑓(𝑥) = −0.02𝑥 2 + 𝑥 + 1 models the yearly growth of a young redwood tree, 𝑓(𝑥), in inches, with x inches of rainfall per year. How many inches of rainfall per year result in maximum tree growth? What is the maximum yearly growth? 4. Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product? 5. A field bordering a straight stream is to be enclosed. The side bordering the steam is not to be fenced. If 1000 yards of fencing is to be used, what are the dimensions of the largest rectangular field that can be fenced? What is the maximum area? 6. Determine the end behavior of each function. a. 𝑓(𝑥) = −𝑥 3 + 𝑥 2 + 2𝑥 c. 𝑓(𝑥) = 𝑥 5 − 5𝑥 3 + 4𝑥 b. 𝑓(𝑥) = 𝑥 6 − 6𝑥 4 + 9𝑥 2 d. 𝑓(𝑥) = −𝑥 4 + 1 7. Find the zeroes for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. a. 𝑓(𝑥) = −2(𝑥 − 1)(𝑥 + 2)2 (𝑥 + 5)3 b. 𝑓(𝑥) = 𝑥 3 − 5𝑥 2 − 25𝑥 + 125 8. Divide using long division. Is the binomial a factor of the given polynomial? a. (4𝑥 3 − 3𝑥 2 − 2𝑥 + 1) ÷ (𝑥 + 1) b. (10𝑥 3 − 26𝑥 2 + 17𝑥 − 13) ÷ (5𝑥 − 3) 9. Divide using synthetic division. Is the binomial a factor of the given polynomial? a. (3𝑥 4 + 11𝑥 3 − 20𝑥 2 + 7𝑥 + 35) ÷ (𝑥 + 5) b. (3𝑥 4 − 2𝑥 2 − 10𝑥) ÷ (𝑥 − 2) 10. Find the end behavior, symmetry, x- and y-intercepts, vertical and horizontal asymptotes, and a table of values for each function. Then, graph the function. a. 𝑓(𝑥) = 4𝑥 − 𝑥 3 2𝑥 b. 𝑓(𝑥) = 𝑥 2 −9 c. 𝑓(𝑥) = 𝑥 2 +2𝑥−3 𝑥−3 11. List all possible rational roots or rational zeros. Then, find the zeros of function. Write the polynomial as a product of linear factors. a. 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 − 4 c. 𝑥 4 − 𝑥 3 − 7𝑥 2 + 𝑥 + 6 = 0 b. 8𝑥 3 − 36𝑥 2 + 46𝑥 − 15 = 0 d. 𝑓(𝑥) = 2𝑥 4 + 𝑥 3 − 9𝑥 2 − 4𝑥 + 4 12. Find an n-th degree polynomial function with real coefficients satisfying the given conditions. a. n=3; 2 and 2-3i are zeros; f(1)=-10 b. n=4; i is a zero; -3 is a zero of multiplicity 2; f(-1)=16 13. Solve the inequality. Express the solution in interval notation and on a number line. 𝑥 b. 5𝑥 ≤ 2 − 3𝑥 2 a. >2 𝑥−1