2. Solve the equation x^2 - 1.6x - 0.23 = 0, obtaining the values of the roots correct to three significant figures. 3. Write out the first four terms of (a - b)^7. Find the fourth term of this expansion when a = [x^(-1) y^(1/2)]^(1/3), b = [9xy^(-4)]^(1/6), expressing the result in terms of a single radical, and without fractional or negative exponents. 4. Reduce the following expression to a polynomial in a and b: (6a^3 + 7ab^2 + 12b^3)/(3a^2 - 5ab - 4b^2) - 1/[3/19b - (5a + 4b)/(19a^2)]. 5. The cost of publishing a book consists of two main items: first, the fixed expense of setting up the type; and, second, the running expenses of presswork, binding, etc., which may be assumed to be proportional to the number of copies. A certain book costs 35 cents a copy if 1000 copies are published at one time, but only 19 cents a copy if 5000 copies are published at one time. Find (_a_) the cost of setting up the type for the book, and (_b_) the cost of presswork, binding, etc., per thousand copies. ~HARVARD UNIVERSITY~ ELEMENTARY ALGEBRA TIME: ONE HOUR AND A HALF 1. Find the highest common factor and the lowest common multiple of the three expressions a^4 - b^4; a^3 + b^3; a^3 + 2a^2 b + 2ab^2 + b^3. 2. Solve the quadratic equation x^2 - 1.6x + 0.3 = 0, computing the value of the larger root correct to three significant figures. 3. In the expression x^2 - 2xy + y^2 - 4[2^(1/2)](x + y) + 8, substitute for x and y the values x = (u + v + 1)/[2^(1/2)], y = (u - v + 1)/[2^(1/2)], and reduce the resulting expression to its simplest form. 4. State and prove the formula for the sum of the first n terms of a geometric progression in which a is the first term and r the constant ratio. 5. A state legislature is to elect a United States senator, a majority of all the votes cast being necessary for a choice. There are three candidates, A, B, and C, and 100 members vote. On the first ballot A has the largest number of votes, receiving 9 more votes than his nearest competitor, B; but he fails of the necessary majority. On the second ballot C's name is withdrawn, and all the members who voted for C now vote for B, whereupon B is elected by a majority of 2. How many votes were cast for each candidate on the first ballot? ~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~ ALGEBRA A TIME: ONE HOUR AND THREE QUARTERS 1. Factor the expressions: x^3 + x^2 = 2x. x^3 + x^2 - 4x - 4. 2. Simplify the expression: [1 - (b^2)/(a^2)][1 - (ab - b^2)/(a^2)](a^4)/(a^3 + b^3) . (a - b)/(a^2 + b^2). 3. Find the value of x + [1 + x^2]^(1/2), when x = (1/2)[[a/b]^(1/2) - [b/a]^(1/2)]. 4. Solve the equations: (7x + 6)/11 + y - 16 = (5x - 13)/2 - (8y - x)/5, 3(3x + 4) = 10y - 15.