BASIC ALGEBRA REVIEW

Assume that letters such as a, b, x, y represent real numbers , unless otherwise specified.

I.
Subsets of the real numbers: Integers include whole numbers and their negatives. The rationals include all
integers AND all fractions, and have repeating decimals . Irrationals (e.g., π ,

, etc.) include all other
real numbers and have non-repeating decimals. A real number is either rational or irrational but not both.

II.
Order: a < b means that a is located to the left of b on the real number line ; a ≤ b means that either a is to
the left of b, or a and b coincide. 0 is greater than all negative numbers, and less than all positive numbers .

III.
Absolute value:

is the distance (on number line) from a to b, disregarding direction (so the result of
an absolute value is always non-negative). If a > b , then

; else

. If
b = 0 , then

is the distance from 0 to a. Watch negative signs closely;

but

IV.
Exponents:

for a ≠ 0 . Change negative exponents to positive by exchanging the position of the
associated factors from numerator to denominator or vice-versa; e.g.,

(note: do not
exchange any factors that initially have positive exponents!). Other rules:

Fractional exponents are really radicals:

V.
Radicals (inverses of exponents): If

then

(read “the cube root of 216 equals 6”; note
that the same three numbers 6, 3, 216 appear in both equations, but in a different order). The small 3
above the radical is the index ; the 216 inside the radical is the radicand. If the index is 2, it is usually not
written; thus

. Simplify radicals when possible by factoring; for a cube root, any group of 3
identical factors is moved outside the radical and compressed into one factor; for fourth roots, any group
of 4 identical factors is moved and compressed, etc.; e.g.,

VI.
Operations involving radicals: Product

; quotient

sum/difference

(similar to combining like terms ); rationalizing denominators (one term)

; rationalizing denominators ( two terms , using conjugate of denominator)

; if an expression
combines different indices (e.g., a cube root times a fourth root), convert radicals to fractional exponents
and use exponent rules to simplify (see IV above); result may be converted back to radicals afterwards.

VII.
Factoring polynomials: Simplify like terms (if any) and rewrite in descending-powers order. Then try
methods in the order listed: (1) GCF ; (2) grouping (for four or more terms); (3) difference of squares, or
sum/difference of cubes (for certain binomials); and (4) trinomial (“ac”) method. It may be necessary to
apply multiple methods (or repeatedly apply a single method) within a single expression, e.g.,

; the difference-of- squares method applies twice.

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