Factors; coefficient; exponent; power; base; term; algebraic sum; similar terms; degree; homogeneous expression; linear equation; root of an equation; root of an expression; identity; conditional equation; prime quantity; highest common factor (H. C. F.); lowest common multiple (L. C. M.); involution; evolution; imaginary number; real number; rational; similar radicals; binomial surd; pure quadratic equation; affected quadratic equation; equation in the quadratic form; simultaneous linear equations; simultaneous quadratic equations; discriminant; symmetrical expression; ratio; proportion; fourth proportional; third proportional; mean proportional; arithmetic progression; geometric progression; S [infinity] ~Special Rules for Multiplication and Division~ 1. Square of the sum of two quantities. (x + y)^2. 2. Square of the difference of two quantities. (x - y)^2. 3. Product of the sum and difference of two quantities. (s + t)(s - t). 4. Product of two binomials having a common term. (x + r)(x + m). 5. Product of two binomials whose corresponding terms are similar. (3x + 2t)(2x - 5t). 6. Square of a polynomial. (m - n/3 + k)^2. 7. Sum of two cubes. (x^3 + y^3)/(x + y) = x^2 - xy + y^2. 8. Difference of two cubes. (x^3 - y^3)/(x - y) = x^2 + xy + y^2. 9. Sum or difference of two like powers. (x^7 + y^7)/(x + y), (x^5 - y^5)/(x - y), (x^4 - y^4)/(x - y), (x^4 - y^4)/(x + y). ~Cases in Factoring~ 1. Common monomial factor. mx + my - mz = m(x + y - z). 2. Trinomial that is a perfect square. x^2 +- 2xy + y^2 = (x +- y)^2. 3. The difference of two squares. (a) Two terms. x^2 - y^2 = (x + y)(x - y). (b) Four terms. x^2 + 2xy + y^2 - m^2 = (x + y + m)(x + y - m). (c) Six terms. x^2 + 2xy + y^2 - p^2 - 2pq - q^2 = [(x + y) + (p + q)][(x + y) - (p + q)]. (d) Incomplete square. x^4 + x^2 y^2 + y^4 = x^4 + 2x^2 y^2 + y^4 - x^2 y^2 = (x^2 + y^2 + xy)(x^2 + y^2 - xy). 4. Trinomial of the form x^2 + bx + c. x^2 - 5x + 6 = (x - 2)(x - 3). 5. Trinomial of the form ax^2 + bx + c. 20x^2 + 7x - 6 = (4x + 3)(5x - 2). 6. Sum or difference of two cubes. See "Special Rules," 7 and 8. two like powers. See "Special Rules," 9. 7. Common polynomial factor. Grouping. t^2 p + t^2 q - 2mp - 2mq = t^2(p + q) - 2m(p + q) = (p + q)(t^2 - 2m). 8. Factor Theorem. x^3 + 17x - 18 = (x - 1)(x^2 + x + 18). ~H. C. F. and L. C. M.~ a^2 + 2a - 3 = (a + 3)(a - 1). a^2 + 7a + 12 = (a + 3)(a + 4). a^4 + 27a = a(a + 3)(a^2 - 3a + 9). H. C. F. = a + 3. L. C. M. = (a + 3)(a - 1)(a + 4)a(a^2 - 3a + 9). ~Fractions~ Reduction to lowest terms.