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Formula Sheet
1 Factoring Formulas
For any real numbers a and b,
(a + b)^2 = a^2 + 2ab + b^2 Square of a Sum (a โ b)^2 = a^2 โ 2 ab + b^2 Square of a Difference a^2 โ b^2 = (a โ b)(a + b) Difference of Squares a^3 โ b^3 = (a โ b)(a^2 + ab + b^2 ) Difference of Cubes a^3 + b^3 = (a + b)(a^2 โ ab + b^2 ) Sum of Cubes
2 Exponentiation Rules
For any real numbers a and b, and any rational numbers
p q
and
r s
ap/q^ ar/s^ = ap/q+r/s^ Product Rule = a
psqs+qr
ap/q ar/s^
= ap/qโr/s^ Quotient Rule
= a
psqsโqr
(ap/q^ )r/s^ = apr/qs^ Power of a Power Rule (ab)p/q^ = ap/q^ bp/q^ Power of a Product Rule ( (^) a b
)p/q
ap/q bp/q^
Power of a Quotient Rule
a^0 = 1 Zero Exponent
aโp/q^ =
ap/q^
Negative Exponents 1 aโp/q^
= ap/q^ Negative Exponents
Remember, there are different notations:
โ qa = a 1 /q โ qap (^) = ap/q (^) = (a 1 /q (^) )p
3 Quadratic Formula
Finally, the quadratic formula: if a, b and c are real numbers, then the quadratic polynomial equation ax^2 + bx + c = 0 (3.1)
has (either one or two) solutions
x =
โb ยฑ
b^2 โ 4 ac 2 a
4 Points and Lines
Given two points in the plane, P = (x 1 , y 1 ), Q = (x 2 , y 2 )
you can obtain the following information:
- The distance between them, d(P, Q) =
(x 2 โ x 1 )^2 + (y 2 โ y 1 )^2.
- The coordinates of the midpoint between them, M =
x 1 + x 2 2
y 1 + y 2 2
- The slope of the line through them, m =
y 2 โ y 1 x 2 โ x 1
rise run
Lines can be represented in three different ways:
Standard Form ax + by = c Slope-Intercept Form y = mx + b Point-Slope Form y โ y 1 = m(x โ x 1 )
where a, b, c are real numbers, m is the slope, b (different from the standard form b) is the y-intercept, and (x 1 , y 1 ) is any fixed point on the line.
5 Circles
A circle, sometimes denoted
, is by definition the set of all points X := (x, y) a fixed distance r, called the radius, from another given point C = (h, k), called the center of the circle,
โ (^) def = {X | d(X, C) = r} (5.1)
Using the distance formula and the square root property, d(X, C) = r โโ d(X, C)^2 = r^2 , we see that this is precisely (^) โ def = {(x, y) | (x โ h) (^2) + (y โ k) (^2) = r (^2) } (5.2)
which gives the familiar equation for a circle.
Theorem 8.3 (Intermediate Value Theorem) Let f (x) be a real polynomial. If there are real numbers a < b such that f (a) and f (b) have opposite signs, i.e. one of the following holds
f (a) < 0 < f (b) f (a) > 0 > f (b)
then there is at least one number c, a < c < b, such that f (c) = 0. That is, f (x) has a root in the interval (a, b).
Theorem 8.4 (Remainder Theorem) If a real polynomial p(x) is divided by (x โ c) with the result that p(x) = (x โ c)q(x) + r
(r is a number, i.e. a degree 0 polynomial, by the division algorithm mentioned above), then
r = p(c)
9 Exponential and Logarithmic Functions
First, the all important correspondence
y = ax^ โโ loga(y) = x (9.1)
which is merely a statement that ax^ and loga(y) are inverses of each other.
Then, we have the rules these functions obey: For all real numbers x and y
ax+y^ = axay^ (9.2)
axโy^ =
ax ay^
a^0 = 1 (9.4)
and for all positive real numbers M and N
loga(M N ) = loga(M ) + loga(N ) (9.5)
loga
M
N
= loga(M ) โ loga(N ) (9.6)
loga(1) = 0 (9.7) loga(M N^ ) = N loga(M ) (9.8)
