Home > Formula Sheet 1 Factoring Formulas 2 Exponentiation Rules

Formula Sheet 1 Factoring Formulas 2 Exponentiation Rules

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Formula Sheet 1 Factoring Formulas
For any real numbers a and b, (a + b)2 = a2 + 2ab + b2 Square of a Sum (a − b)2 = a2 − 2ab + b2 Square of a Difference a2 − b2 = (a − b)(a + b) Difference of Squares a3 − b3 = (a − b)(a2 + ab + b2) Difference of Cubes a3 + b3 = (a + b)(a2 − ab + b2) Sum of Cubes
2 Exponentiation Rules
For any real numbers a and b, and any rational numbers p q and r s , ap/qar/s = ap/q+r/s Product Rule = a
ps+qr qs
ap/q ar/s = ap/q−r/s Quotient Rule = a
ps−qr qs
(ap/q)r/s = apr/qs Power of a Power Rule (ab)p/q = ap/qbp/q Power of a Product Rule (a b )p/q = ap/q bp/q Power of a Quotient Rule a0 = 1 Zero Exponent a−p/q = 1 ap/q Negative Exponents 1 a−p/q = ap/q Negative Exponents Remember, there are different notations:
q
�� a = a1/q
q
�� ap = ap/q = (a1/q)p 1

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3 Quadratic Formula
Finally, the quadratic formula: if a, b and c are real numbers, then the quadratic polynomial equation ax2 + bx + c = 0 (3.1) has (either one or two) solutions x = −b �� �� b2 − 4ac 2a (3.2)
4 Points and Lines
Given two points in the plane, P = (x1,y1), Q = (x2,y2) you can obtain the following information: 1. The distance between them, d(P, Q) = ��(x2 − x1)2 + (y2 − y1)2. 2. The coordinates of the midpoint between them, M = (x1 + x2 2 , y1 + y2 2 ) . 3. The slope of the line through them, m = y2 − y1 x2 − x1 = 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 − y1 = m(x − x1) where a, b, c are real numbers, m is the slope, b (different from the standard form b) is the y-intercept, and (x1,y1) 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, �� def = {X | d(X, C) = r} (5.1) Using the distance formula and the square root property, d(X, C) = r ⇐⇒ d(X, C)2 = r2, we see that this is precisely �� def = {(x, y) | (x − h)2 + (y − k)2 = r2} (5.2) which gives the familiar equation for a circle. 2

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6 Functions
If A and B are subsets of the real numbers R and f : A �� B is a function, then the average rate of change of f as x varies between x1 and x2 is the quotient average rate of change = ∆y ∆x = y2 − y1 x2 − x1 = f(x2) − f(x1) x2 − x1 (6.1) It��s a linear approximation of the behavior of f between the points x1 and x2.
7 Quadratic Functions
The quadratic function (aka the parabola function or the square function) f(x) = ax2 + bx + c (7.1) can always be written in the form f(x) = a(x − h)2 + k (7.2) where V = (h, k) is the coordinate of the vertex of the parabola, and further V = (h, k) = ( − b 2a ,f ( − b 2a )) (7.3) That is h = − b
2a
and k = f(− b
2a
).
8 Polynomial Division
Here are the theorems you need to know: Theorem 8.1 (Division Algorithm) Let p(x) and d(x) be any two nonzero real polynomials. There there exist unique polynomials q(x) and r(x) such that p(x) = d(x)q(x) + r(x) or p(x) d(x) = q(x) + r(x) d(x) where 0 �� deg(r(x)) < deg(d(x)) Here p(x) is called the dividend, d(x) the divisor, q(x) the quotient, and r(x) the remainder. �� Theorem 8.2 (Rational Zeros Theorem) Let f(x) = anx2 + an−1xn−1 + ··· + a1x + a0 be a real polynomial with integer coefficients ai (that is ai �� Z). If a rational number p/q is a root, or zero, of f(x), then p divides a0 and q divides an �� 3

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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 − c) with the result that p(x)=(x − 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 ⇐⇒ 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) ax−y = ax ay (9.3) a0 = 1 (9.4) and for all positive real numbers M and N loga(MN) = loga(M) + loga(N) (9.5) loga (M N ) = loga(M) − loga(N) (9.6) loga(1) = 0 (9.7) loga(MN ) = N loga(M) (9.8) 4

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