Ifthe signs ofthe numbers are different, subtract.. Multiplication: Multiply the numbers, then detcrmine the sign of the answer.. Remember: negative· negative = positive; positive· posit
Trang 1For any real numbers a b, and c:
A,Closure
I For addition: a + b is a real number
PROPERTIES OF INEQUALITY
For any real numbers a, , and c:
A Trichotomy: Either a = b, a < b or a > b
B Transitive: If a < band b < c then a < c; also, if a> band b > c, then a > c
2 For 1l1ultiplication: a • b is a real number
B, Commutative Property
I For addition: a + b = b + a
2 For multiplication: ab = ba
C Associative Property
I For addition: a + (b + c) = (a + b) + C
2 For multiplication: a(bc) = (ab)c
D Identity Property
I For addition: a + 0 = a and 0 + a = a
2 For multiplication: a· 1 = a and I • a = a
E I nverse Property
I For addition: a + (-a) = 0 and (-a) + a = 0
2 For multiplication: a· t = I and t a = I
F Distributive Property
a(b + c) = ab + ac; a(b - e) = ab - ac; and ab + ac = a(b + c); ab - ac = a(b - c)
G Multiplication Property of Zero
o • a = 0 and a • 0 = 0
H Double Negative Property
-(-a) = a or -1(-1· a) = a
Ixl = x if x ,,() and -x if x < O It is always a positive numerical value
Ixl = I-xl
3 Ixl" 0
4 Ix - yl = Iy -xl
B Addition: If the signs of the numbers are the same, add Ifthe signs ofthe numbers
are different, subtract In both cases, the answer has the sign of the number with
the larger absolute value
C Subtraction: Change subtraction to addition of the opposite number, then follow
the addition rules
D Multiplication: Multiply the numbers, then detcrmine the sign of the answer
Remember: negative· negative = positive; positive· positive = positive;
negative· positive = negative; positive· negative = negative; if the signs are the
same, the answer is positive; if signs are different, the answer is negative
E Division: Divide the numbers, then determine the sign ofthe answer using the same
sign rules that apply to multiplication
OPERATIONS OF COMPLEX NUMBERS
A Definition: a ± bi where a,b E Real numbers and i = FI
B Addition: (a ± bi) + (c ± di) = (a + c) + (b ± d)i ,
C Subtraction: (a ± bi) -(e ± di) = (a - c) ± (b =+= d)i
D Multiplication: (a ± bi)(c ± dt) = (ac =+= bd) ± (ad ± bc)i
E Division:
a±bi a±bi c =+= di (ac=+=bd) ± (ad±bc)i
c±di = c±di' c=+=dj = c'±d'
C.Addition Property: If a < b, then a + c < b + c; also, ira> b, then a + c > b + c for any value of e
D Multiplication Property: If c > 0 and a < b then ae < bc If C > II ad a > b, then ae > be Ifc <()and a < b, then ac> be.lfc <0 and a > b, then ae < be Ife = ()then ac = bc =O
OPERATIONS OF ALGEBRAIC EXPRESSIONS
A Like or similar terms are terms with the same vriables hving the samc exponent values
ADDITION/SUBTRACTION OF POLYNOMIALS
ax + bx = (a + b)x or ax - bx = (a - b)x; if the variables and exponents arc the same add or subtract the numbers in front (coctficients) without changing the variable
I x ± x = -x ; If the denol11111ators arc the same, add the numerators only
2 x a ± y b = xy ± ay bx yx = -xy; ay ± bx " It t lC (enomillators arc not ·1 I ' Il C same, multiply each fraction by one in the required form, then add the numerators If the denominators arc polynomial, then factoring em fir wi II h lp determine the least common denominator
MULTIPLICATION RULES
I x· x • x x = x" when the number of x vari bles = n
2 xO = 1 when x 0
3 Xl = x
4 xm x" = xm + ; also axlll • bx" = abx'" + ; multiply coefficients and variables
S (Xlll)" = xm"; also (Xmyl')" = X"H'yP"; powers of powers of monomials can be
done by mu Itip lying exponents
6 Y = ym
7 II' a" = a V then 1/ = v
8 If a" = b" for a 0, then a = b
DISTRI BUTIONS
I Type I: a(x + y) = ax + ay; a(x - y) = ax -ay
2 Type 2:
a (a + b)(x + y) = a(x + y) + b(x + y) = ax + ay + bx + by
b This type, 2 terms times 2 terms, an also be cione using the FOIL rule: First term times first tcrm, Outer term timcs o ter term, Inner term times inner term, Last term timcs last term, (a + b)(x + y)
3 Type 3: (a + b)(x + Y+ z) = a(x + y + z) + b(x + Y+ z) = ax + ay + az + bx + by + bz
4 Thc Binomial Theorcm: The expansion of(x+y)", where n is a countng number,
is a.x" + a2x n-1y + a)xn-Zy2+.•.+ an_txyn-I + anyn when: a l " a2• 3y '0', an arc found in Pascal's triangle:
I I
3
5 10 10 5 I and so on
S The BII101111al Formula: (x+y)" = x" + T x"- y + I, 2 x" y-+ +
A Reflexive: a = a
B Symmetric: If a = b, then b = a
C Transitive: I f a = band b = e, then a = e
D Addition Property: If a = b, then a + e = b + C for any value of c
E Multiplication Property: If a = b, then ac = bc for any value of c
F P ropor t' IOn P roper y: t It' 11 = ct' t a c h en a = d b c; C a = ct; b a b = c d : a c 11 d
t ; -d-; a + b c + d
1
n(n-1)(n 2) (n - r+ I) nor r II
I ' 2 ' 3 r x y + + Y
6 The Binomial Coefficient: [II] = I ( I~ ) ,
r r n r
7 Othcr special types of distributions arc listed under the Factors secton
I 11'ct=bd
2 polynomial polynomial
RATIONAL EXPRESSIONS
I I l11 ust ha e all 01 the nUlllcrators and dcnOllllllutorS
po ynolllla factored so identical factors (one in a numerator und the other in a denomi nator) can be cancelled before l11ultirlying
Trang 2DIVISION FIRST DEGREE WITH ONE VARIABLE· STEPS
and not zero, the fraction equals one
ax
2 ay = )7; common factors cancel to equal one
x III _ 111-11 , X III _ 1 ,
3 - I I - x II 111 > nand -n - ~ 11 m < It
4 b a ~ d c a d = b • c = ad he; I · any ·1' 0 t' t le numerators I or enol1lmators d ' are po ynoI
2
mials, like ax 2 ± b, ± e they must first be factored so identical factors can
dx ± ex ± f
be cancelled as indicated in the Multiplication Rules section
b a e a d I f" d· d d'
5 C = b -7 d = b • c : comp ex ractlons are In Icate ,vIsion
d
T ± l l fil ± Tt1 - rh
I'h (ad ±
bd (eh ±
NEGATIVE EXPONENTS
J X-III = -k and -h= xln
2 (x ± yrm = I '" and '" = (x ± y)"'
FRACTION EXPONENTS/RADICAL EXPRESSIONS
I a ~ = rVaW and IVaW = a*
I
2 'V;; = a /I where b" = a; ir n is even and a < 0 'V;; is not a real number,
and if n is even and a " 0, 'V;; is the nonnegative number b
3 'Va" = a; (a" )+.= a * = a' =a ifn is ml and Ia Iifn is even, and ( a+' r= a *=a' =a
4 (ax)7 = ·V(ax) lH= nja ll1 Xlll = rv;m ' R;and ,v;m 'R= IVa III xlll=(ax)*
5 (ax)~ = 1(ax)+']", = ,~ ; and 'V(ax)'" = l(ax)+'1'" = (ax)~
6 (ax ± by) ~ = 'V(ax ± by)'" BUT it docs NOT equal 'v(ax)'" ± 'V(by)'"
7 (XII')' )+' = ,,~ = '-':; = x:: BUT if the expression is to be written in
y Vy I' It' YP YII
S 'R =" ~ = (~ ,)·,"
FACTORS
I Linear Factor Theorem: A polynomial of degree /I " I can be written as th
product of /I linear l~lctors, P(x) = a(x - I",)(x - 1"2)"'(X -r
2 Greatest common factor: ax ± ay = a(x ± y)
3 Quadratic trinomials:
a Xl ± bx ± c = (x + h)(x + k) where h • k = c and h + k = b hand k can be
b ax l ± bx ± c = (mx + h)(nx + k) where a " I and m· n = a h • k = c
m • k + h • n = b; m n, h, and k can be positive or negative numbers
4 Perfect square: Xl ± 2xy + yl = (x ± y)(x ± y) = (X ± y)l
5 Perfect cube: x3 ± 3xly + 3xyl ± y3 = (x ± y)3
6 Difference of two squares: Xl -yi = Xl + xy - xy -; i = (x + y)(x -y)
7 Sum of two squares: Xl + y l = (x + YI)(x - YI), where;", / - I and is an imaginary number
8 Difference of two cubes: x.1 - Y" = (x - Y)(XI + '1 + yl)
9 Sum of two cubes: x3 + y.1 = (x + Y)(XI - xy + y )
10 Grouping
a.2 -2 grouping: ax ± ay ± bx ± by = a(x ± y) ± b(x ± y) = (a ± b)(x ± y)
c I -3 grouping yl -Xl ± 2ex - cl = yl -(x ± C)l = (y + x ± c)(y - x + c)
I I Partial-fraction decomposition rules
a Linear factors: For each distinct factor of the form (ax + h)1II in the denom
inator Q(x) , introduce the sum of 11/ partial fractions A+I b + A2
+ + (ax + b)'" w lere I' I' , m are constants
denominator Q(x), introduce the sum of m partial fractions A I X + B I +
ax 2 + bx + e
(ax2+bx+c)2 (ax2+bx+c)lI1were I' 2' ••• ' man l' 2'
, B are constants
2
number was used to distribute throughout the entire inequality, then the inequality
SYSTEMS OF EQUATIONS
I A consistent system has one or more solutions
2 An inconsistent system has no solution
3 Methods of solution
on the right side of the equals sign)
2)Distribute through one or both cqua ons so the coelTicicnts (nul11bers in
4)Solve the resulting cquation for the remaining varia
2)Put the rcsulting statement into the other equation in place of the variable
(the resulting equation will have only one vari bl
3)Solve the equation for the numc cal value of the variable
4)Put this numerical value inro either of the two beginning equations and
solve for the second variable numerical alue
c Graphin
I )(jraph each equation on the coorc!inat pla
2)Find and label the point(s) of intersection, if there
d Cramer's Rule
I )Put both equations in standard form (all habetica order with the constant
on the right side of the equals sign): ax + by = c and dx + ey = f 2)bke the coefficients (numbers in fi'ont of the v riables) and make deter minate of the system of equations J) = ,~ ~,
is the determinate of that variable), D , = , ~: ~,
4)Take the original determinate of the systel1l and replace the coeff ients
of the other v riable with the constants of the system (this is the deter minate of the other variable), J) , = I ~ ~I·
5 )Solvc each deterl11inate hy finding th ~ dilTercnce 0 the cross product
6)The solutions are x = ~' and y ~
I A matrix is a rectangular array of real numbers, called entries or clements
al l a l 2 al II
ami a 1112 i lrnn columns and 11/ = number of rows
2 The dimension is " 11/ by /I "
3 A coefficient matrix is formed by the c e icients of one variable of a system
of linear cquations forming each column of the matrix
4 An augmented matrix includes the constants of a system of lincar equations separated by a vertical ashed line in the matrix
5 Row operations that transform an augl11cntcc!lllatrix into an equivalent system:
a Interchange any twO roWS
b Multiply every clcmcnt of any row by a consta t, C , where c o
c Replace every clement of any row by the slim of iself and a corre
matrix until all clements of the matrix, xccpt the constants are zeros and
I () 0: e j
() () I, ' C J
Trang 3clements
of the matrices
the matrix by the scalar c
6)/\ determinant is a number c lculated Ii·om a square matrix (matrix with
by (_ l) i + j
SECOND DEGREE·QUADRATIC
a
i Usc inverse operations to set the equation equal to zero
i
jv Solve each resulting
2)Quadratic Formula
I Usc inverse operations to set the equation equal to zero
the constant without a variablc
2 Polynomials
degree II " I with rcal coetTicients then a -bi is also a root
of PC-x»~ or is less than the number of variations in sign by an even number
a Isolate the radical expression on one side of the equation, if possible (if not
possible, put the radical expressions on two opposite sides of the equals sign)
B Each point is named by an orde"cd pair, (x, y)
C The dista ce between two points is d = j' ;x - _ - x- ,)-'-+-(-y-, -y-,-)'
D The midpoint ofa line segme t with endpoints (x " Y ,) and (xz' Yz) is P(.>: , y) where
x= - -2 -an y = - - 2 - '
E A relation is a set of ordered pair
~~ A function is a relation that has no x-values that are the
I j(x) is read "fofx " or "the fll l/cti o l/ (jrx"
2 (r + g)(-.:) =f(x ) +g (x)
4 (rg)( x) = /(.>.:) • g(.!;)
S (r/ g )(!;) =j(x ) g ( :)
C Vertical line test of a function: A g raph represents a function if and only if no
H A one-to-one function is a function that has no y-valucs that are thc sal1lc
POLYNOMIALS
I Form:.f(x) = anx n + an_lxn-t+ +a tx+ao for real numbers; a, with an '" O
2 Restric o s on the coordinate values may bc nccessary for the polynol1lial to be
a function
.f(a) "'j(h) , thenf takes every value bctwccnf(a) and.f(b) in the interval la, b J
S Inverse functi ons,f -'( :), are found by cxchanging the x and the)' variables in
y in the range
7 Factor Theorem: The poly omial P( :) has a factor x -, if and only if P ( r = 0
and r is a root
~ If the c efficients of the polynomial P(x) = anx" + an_1x"-'+ +a1x+aoarc
integers and * is a rational root in lowest terms, then p is a factor of the
r is real root of Q(x) but not of Pix)
10 The tests for symmetry of graphs symmetric with respect to the
a x-a s i replacin y in the equation with -)I results in an equivalent equation
b.y-axis if replacingx in the equation with - x results in an cquiHtient equation
c origin ifrcplaein both x with -x andy with -)' in the equation rcsults in an
equivalent equa on
EXPONENTIAL FUNCTION
I Fonn:j(x) = a " , where a > 0 I I '" I
2 Inverses of e ponential fu ctio s arc logarithmic functions
LOGARITHMIC FUNCTIONS
2 The common logarithm, lo x , has a base of 1 so II = 10 in the definition ofl og
4 Propertes with II > 0 and II '" I
a al oga x = x
f If log"u = logbu a d " I, then a = b
g.log"xy = logax + log,,)"
h.loga ( f ) ~ log"x -logY
j logax" = n(log"x), where n is a real number k.Change of base rule: If a > 0 a I, b > 0 b '" I, and x > 0 then
( log x)
I In x ( log e)
3
Trang 4CONIC SECTIONS
The general limn of the equation ofa conic section with axes parallel to the coordinate
axes is Ax2 + B.W + CJl + Dx + Ey + F = 0, where A and C are not both zero,
a Slope = III = X I ~ X 2 = ~= riiIl
b Point-slope form 01' a linea~ equation: y - y / = III(X - x I)'
e Slope-intercept form of a linear equation: y = IIIX + b, where 111 is the slope
d Standard form ofa linear equation: ax + by = c, where a, b, and c are integers
g Two lines arc parallel ifand only if their slopes are equal; m l = m 2
rocals, and m 2 = - nil and m l = - m,'
i If III > 0, the line is increasing
j If 111 < 0 the line is decreasing
k If 111 = 0 the line is horizontal
1 If III is undefined, the line is vertical
a General equation: y = a(x - h;2 + k (opens up/down), or x = a(I' - k)2 + h
(opens lelilright)
(opens Icft/right)
c Where (h k) is the vertcx (h k ± p) or (h ± P k) is the vertex with p = 41a'
y = k ±porx = h ±pis the directrix andx = h ory = k is the line ofsymmctry
a General equation: (x - h)1 + (y - k)1 =,2
b Where (I! k) is the center of the circle and r is the radius
a General equation: - - , - + - - , -= I
the center to points on the ellipse and b is the vertical movement up and
e Additionally, when a > b, then major axis is horizontal and foci are (h ± c
k); where c2 = {/1 -b 2 and where a < b, then major axis is vertical and foci
are (It k ± c), where ("1 = b 2 _ a 2
(x - h) (y - k) l ' I f ' h)
a enera equatIOn: - - , - - - - , -= (opens et-ng t
b Where (It k) is the cente~of the rectaRglc that contains no points ofthe hyperbola,
a is the horizontal movement left and right ll'Om the center to the points on the
rectangle, b is the vertical movement up and down /Tom the center to points on the
(y'~k)' (x-h)'
c General equation: - -, - - - , - = 1 (opens up-down)
d Where (It k) IS the center of the rectangle that contams no pomts of the hyperbola,
b is the horizontal movement left and right from the center to the points on the
SEQUENCES & SERIES
written as a I' a2' a3' ••• , ai' , with each ai representing a term
B A finite sequence is a function with a domain that is a set or only II positive
integers; written as U ,'"r "3' , (I,,_i' a/,
l "" I
D.lllh partial sum: S" = :t a, = a l + a 2 + + a"'1 + a"
l = 1
difference; a" = a".1 + d where d is the common difference
f A geometric sequence or geometric progression is a sequence in which each term
is a constant multiple of the preceding term; an = ran_I where r is the constant
G II! or " II factorial" = 11(11 - /)(11 - 2)(11 -3) (3)(2)(/); note: O! = t
PROPERTIES OF SUMS, SEQUENCES & SERIES:
I :t(a, + b.) = :tal ± :tb,
3 :te = ne where c is a constant
,,= I
common difJerence
5 Arithmetic Series: The sum of the first II terms of an arithmetic sequcncc with
a / as the first term and" as the common difference is the III" partial sum:
S n 2 1 n =.!l (a + a ) or S =.!l 12a +(n - 1)dln 2 I
6 The IIlh term of a geometric sequence with a/as the first term and r as thc common ratio is an = alr"-I
called the geometric means
r
term and r as the common ratio, where Ir I ~ I is I ~ r ; when Ir I< I because if Ir I> I or Ir I = I the sum docs not exist
10 The rlh term of the binomial cxpansion of (x + V)" is
n! x I 1 - ( r - l ) r - I
PROBLEM SOLVING
ODD NUMBERS, EVEN NUMBERS, MULTIPLES
1 First number = x
2 Second number = x + d
3 Third number = x + 2d, etc where d is the common di flcrcnee between any tw
MONEY, PURCHASES
3 Formula: VIC I +V 2C 2 + +V"C" =Vlnlal
MIXTURE, SOLUTIONS
3 Formula: VIP I + V 2P2 + + V,llI = Vn"al v"lu.Pfinal valuc
WORK
compared to the time to complete the job alone by one person or o c ma hinc
2 One, t, represents the whole job
3 Formula: WI +W2 +",+Wn = I
DISTANCE
I d is distance
2 r is rate or speed
3 t is time as indicated in the rate; for example miles per hour or meters per second
4 Formula: d = rt
5 These relationships can be used d pending on the situation described in the problem:
a d lo = d ,lurning
b d l + d 2 = dllll"1
1 (I b c and d are quantities specified in the problem
2 k", 0
a c
b Direct varialion: y = kx
d Combined variation:)I = k~x; y varies directly as x and inversely as :
" li te : I)u :: I II ils ", )[1J~ · n d Il' mn:ll u sc IhJ~ Q U i ckStud y
ISBN-13: 978-157222721-7 : han ;1:; <I gujJc hu! no! ;'" a rcplacmcnl I\)f lhlgtlo!J das,\.lo\)rl ISBN-1D: 157222721-4
\U ri l: h r~f l"\ t d "'( , ~lIn h Ih" f"Jh1It.lh.'" , a,n ht ~rd '" Ir~n,,"lH1 :d If! 1111'0' h· m, (If ~ ~,,\ II'\.: J'" ,Irtln.n Of" rm:d\aIU"'.lI "l\:ludll', flh(>tQo,'('fI\ ' ,j,ll' "I 111 m'<lnn~lI"n
""', ' lte dn,J ""r , :\~ 1 '~"'tm ",Iho"" "WI,m pnTll" ,.lll Ir~1In lilt
put-ll_her C2 0 02 2005 B arC h 8r1a, I nc , 04 01
fr~~ ~ r ~akO ~fn t fe s at
qUlcKsluay.com
4