Part 3

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.

Reduction of a mixed number to an improper fraction.

Reduction of an improper fraction to a mixed number.

Addition and subtraction of fractions.

Multiplication and division of fractions.

Law of signs in division, changing signs of factors, etc.