1. Review the Binomial Theorem laws. (See Involution.) Expand: 2. (b - n)^7. 3. (x + x^(-1))^5. 4. [a/x - x/a]^6. 5. [x/2y - [xy]^(1/2)]^5. 6. (x^2 - x + 2)^3. 7. [(2[b^2]^(1/3))/(y) + (3[y^(1/2)])/(b^3)]^4. 8. (a + b)^n = a^n + na^(n - 1)b + [n(n - 1)]/(1.2) a^(n - 2)b^2 + [n(n - 1)(n - 2)]/(1.2.3) a^(n - 3)b^3 + [n(n - 1)(n - 2)(n - 3)]/(1.2.3.4) a^(n - 4) b^4 + .... Show by observation that the formula for the (r + 1)th term = [n(n - 1)(n - 2)...(n - r + 1)]/[1.2.3.4 ... r] a^(n - r)b^r. 9. Indicate what the 97th term of (a + b)^n would be. 10. Using the expansion of (a + b)^n in (8), derive a formula for the rth term by observing how each term is made up, then generalizing. Using either the formula in (8) or (10), whichever you are familiar with, find: 11. The 4th term of [a + 1/a]^(30). 12. The 8th term of (1 + x[y^(1/2)])^(13). 13. The middle term of (2a^(3/4) - y[a^(1/3)])^(10). 14. The term not containing x in [x^3 - 2/x]^(12). 15. The term containing x^(18) in [x^2 - a/x]^(15). ~Reference:~ The chapter on The Binomial Theorem in any algebra. ~MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND~ 1. Solve the equation x^2 - 1.6x - .23 = 0, obtaining the values of the roots correct to three significant figures. (_Harvard._) 2. Write the roots of (x^2 + 2x)(x^2 - 2x - 3)(x^2 - x + 1) = 0. (_Sheffield Scientific School._) 3. Solve 2[2x + 2]^(1/2) + [2x + 1]^(1/2) = (12x + 4)/([8x + 8]^{1/2}). (_Yale._) 4. Solve the equation V = (H/3)(B + x + [Bx]^(1/2)) for x, taking H = 6, B = 8, and V = 28; and verify your result. (_Harvard._) 5. Solve { x : y = 2 : 3, { x^2 + y^2 = 5(x + y) + 2. 6. Solve 2x^2 - 4x + 3[x^2 - 2x + 6]^(1/2) = 15. (_Coll. Ent. Board._) 7. Find all values of x and y which satisfy the equations: { x^(1/2) + y^(1/2) = 4, { 1/[[x + 1]^(1/2) - x^(1/2)] - 1/[[x + 1]^(1/2) + x^(1/2)] = y. (_Mass. Inst. of Technology._) 8. If [alpha] and [beta] represent the roots of px^2 + qx + r = 0, find [alpha] + [beta], [alpha] - [beta], and [alpha][beta] in terms of p, q, and r. (_Princeton._) 9. Form the equation whose roots are 2 + [3]^(1/2) and 2 - [-3]^(1/2). 10. Determine, without solving, the character of the roots of 9x^2 - 24x + 16 = 0. (_College Entrance Board._) 11. If a : b = c : d, prove that a + b : c + d = [a^2 + b^2]^(1/2) : [c^2 + d^2]^(1/2). (_College Entrance Board._) 12. Given a : b = c : d. Prove that a^2 + b^2 : (a^3)/(a + b) = c^2 + d^2 : (c^3)/(c + d). (_Sheffield._)