(Easy)Find the multiplicity of the given zero in the polynomial. p(x) = (x + 1)(x − 3); c = −1 p(x) = (x − 2)3(x − 5); c = 2 p(x) = x2 − 2x − 8; c = 4 p(x) = x2 − 4x + 4; c = 2 (Easy)Find a polynomial with the given conditions. (Answers may vary) degree 3, zeros -2, 0, 3 degree 3, zeros 0 (multiplicity 2) and 1 (Easy)Find the unique polynomial with the given conditions. degree 2, -3 and -1, leading coefficient an = −3 degree 3, zeros 2 (multiplicity 2) and -4, p(3) = 21 (Easy)Divide p(x) by d(x) using long division. p(x) = 6x4 − 4x3 + 3x2 − 14x + 2; d(x) = 3x − 2 p(x) = 2x3 − 5x2 + 1; d(x) = x2 − 3 p(x) = 10x3 + x2 + 18x + 2; d(x) = 2x2 − x + 4 (Medium)Use synthetic division to do the following for p(x) and d(x) = x − c: Divide p(x) by d(x) = x − c. Rewrite p(x) in d(x)q(x) + r(x) form. Evaluate p(x) at x = c. p(x) = 2x4 − 3x3 − 5x2 + x − 6; d(x) = x − 1 p(x) = 3x4 − 2x3 − 4x + 8; d(x) = x + 2 p(x) = x3 − 7x2 + 17x − 20; d(x) = x − 4
Find a polynomial with real coefficients that satisfies the conditions. (Easy)degree 4; zeros 1, -2, 3i (Easy/Medium) degree 5; 2 (mult. 2), -1, -1+2i degree 3; zeros 0, i; p(1) = 4 degree 4; -1 (mult. 2), 1-i; p(2) = −18 Given a zero of the polynomial, factor the polynomial into linear factors. (Medium) p(x) = x3 − 3x2 + 4x − 2; 1 (Medium) p(x) = x3 + 5x2 + 17x + 13; -1 (Hard) p(x) = x4 − 6x3 + 14x2 − 24x + 40; 2i (Hard) p(x) = x4 + 2x3 + 10x2 + 24x + 80; 1 + 3i (Hard)Do the following for each polynomial: Factor into linear and irreducible quadratic factors. Completely factor into linear factors. p(x) = x4 − 3x3 + 26x2 − 22x − 52 p(x) = x5 − x4 − 4x3 − 4x2 − 5x − 3 For each of the following, give an equation for each asymptote. Sketch the graph using the asymptotes and x-intercept(s), if any. (Easy) (Medium) (Medium) (Medium) (Medium) (Medium) (Hard) (Hard)
(Medium) Radium-226 has a half life of 1600 years. How long will it take 5 grams of radium- 226 to be reduced to 2 grams? Use the model , where m is the amount of radium-226 after t years, m0 is the initial amount of radium-226, and k is the decay rate. (Medium) An apple pie is taken out of the oven with an internal temperature of 325oF. It is placed on a rack in a room with a temperature of 72oF. After 10 minutes the temperature of the pie is 200oF. What will be the temperature of the pie 30 minutes after coming out of the oven? Use Newton’s law of cooling: , where Ts is the temperature of the surroundings, D0 is the initial temperature difference between the surroundings and the object, and t is time. (Medium/Hard) The number of trout in a particular lake is given by , where t is time in years. How long will it take for the population to reach 5000? Systems of linear Equations and Inequalities Solve the following systems of linear equations by substitution: (Easy) (Easy) 4x − 5y = −7 3x + 8y = 30 (Medium) the following systems of linear equations by using elimination: (Easy) 2x + 5y = 5 −4x − 10y = −10 (Easy) 3x − 2y = 12 4x + 3y = 16 (Medium) −0.5x + 0.3y = 0.8 −1.5x + 0.9y = 2.4 (Easy) Graph the system of equations to solve: 2x + y = 3 2x + y = 7 x − 2y = 1 2x − 4y = 2
(Easy) (Medium) (Medium) Find the partial fraction decomposition for each rational function: (Easy) (Easy) (Medium) (Medium) (Hard) (Easy) Graph the linear inequalities: y < 2x + 3 5x + 3y < 15 6x − 3y ≥ 9 Graph the system of inequalities or indicate that the system has no solution: (Easy) y > 2x + 1 y < 2x − 1 (Easy) x + 2y > 4 y < 1 x ≥ 0 (Easy) y < x + 2 y > x − 2 y < −x + 2 y > −x − 2 (Medium) y + x < 2 y + x ≥ 4 y ≥ −2 y ≤ 1