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© Copyright 2003, Jim Loy
Under construction. Warning: fines double in construction zone.
Variables and equations: See The Essence of Algebra and Word Problems.
Precedence: What is ab+c? Is it (ab)+c or a(b+c)? Note that we always do what is inside the innermost parentheses first. It is (ab)+c. How about a+bc? OK, that is a+(bc), multiplication has precendence over addition, so we multiply first. How about a+b-c? Addition and subtraction have the same precendence, and so we just go left to right: (a+b)-c. If there can be any doubt, use parentheses to clarify. Powers have precedence over multiplication. In grammar, we have many kinds of parentheses: {[()]} etc. In algebra, we can use the same kind over and over, but they come in pairs which must be matched starting with the outermost pair and working inward: (a+(b(c+d))-b).
Division by zero: Division by zero is illegal. It is not infinity, it is undefined. See Two Equals One?
Rules of algebra:
We can combine these rules in many ways.
Distributive law: a(b+c) = ab + ac. An application of this is that 3x+7x = 10x. Here we used (b+c)a = ba + ca, which you should realize is also the distributive law.
Simplification: We often use the rules of algebra to simplify expressions and equations. Sometimes we need to complicate part of an expression in order to simplify the larger expression (now or later). 2+4 simplifies to 6.
Fractions: a/b + c/d = (ad+bc)/bd. a/b . c/d = ac/bd. I will use "." for multiplication here. (a/b) / (c/d) = ad/bc, where c is not zero. 1/a is called the reciprocal of a. Decimals (like 1.328) are fractions: 1.328 = 1328/1000. Some fractions are repeated decimals: 1/6 = 0.166666... or 0.16 with a line over the 6 to show that it repeats. Fractions should often be reduced to their lowest terms (see Euclid's Algorithm). Cancelling is division (or subtraction): 3x/3 = x (we cancelled the threes, which means that we divided both top and bottom by 3).
Powers: a^b is a standard way to type a to the b power. (x^4) . (x^3) = x^7 (we add the powers). (x^7)/(x^3) = x^4 (we subtract the powers). x^4^3
Polynomials: 3x^3 + 4x^2 + 2x + 9 is a 3rd power polynomial. We can add and subtract polynomials: (2x^2+3x-1) + (3x^2-x+2) = 5x^2 + 2x +1. This is easier to do in columns, with the x^2 terms in one column, the x terms in the next, etc. This is just an application of the rules of algebra. When we multiply polynomials, we multiply the various combinations of terms: (3x+7)(2x^2-4x-1) = 6x^3+14x^2-12x^2-28x-3x-7 = 6x^3+2x^2-31x-7. This bears a resemblance to the way we multiply numbers. Some easy products: (a+b)(a-b) = a^2 - b^2, and (a+b)^2 = a^2+2ab+b^2, and (a-b)^2 = a^2-2ab+b^2. Also see Synthetic Division and Quadratic Formula.
Factoring: We factor a number like 6 = 2 . 3. Similarly 3x+6y = 3(x+2y), an application of the distributive law. x^2-4x+4 can be factored into (x-2)^2. Many polynomials cannot be factored into terms with whole numbers.