Balik Aral (Algebra)
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Transcript of Balik Aral (Algebra)
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1st Balik-Aral: Academic Review Program 1
CHAPTER I: INTEGER EXPONENTS
For any real number a, with an integer exponent n, the nth power of a is an.
•
If n is a positive integer, then an
= a⋅a⋅a… a (n times).• -an and (-a)n are not always eual.
-!" = -!#!#!#! = -1$
(-!)" = (-!)(-!)(-!)(-!) = 1$
• If a % &, then a& = 1.
• If n ' & and a % &, then a-n =1
n.
*+ F /00+
• an # am = an+m xample2 22⋅2
4=22+4=2
6
x 3 # x -4 = x -" =1
x
4
• (an) m = anm xample2 (53)! = 51&
• (ab)n = anbn xample2 (-!⋅5)" = (-!)"⋅(5)"
x 6 # 6 = ( x )6
• If a % &, thena
n
am
= an-m xample278
74
= 67-" = 6"
y3
y−10
= y3−(−10)= y
13
• If b % &, then ( a
b )n
=a
n
bn
xample2 ( x
y )2
= x
2
y2
5
3
153=(
5
15 )3
= 1
27
8/+
•
36 x6 y
3 z
7
32 x
5 y
9 z
7 =
22⋅3
2
32
x6−5
y3−9
z7−7
¿4 x1 y−6
z0
¿ 4 x
y6
(−5 x3 y
−4)4( x5 y
2)−5=(−5)4 x3(4 )
y−4 (4 )
x5(−5)
y2(−5)
¿625 x12
y−16
x−25
y−10
¿ 625
x13
y26
•
x−1+ y
−1
( x+ y )−1=
1
x+ 1
y
1
x+ y
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1st Balik-Aral: Academic Review Program 5
o multiply polynomials, apply the *#st"#!+t#, law o- '+lt#&l#$at#on o,"
a**#t#on and the laws o- %&onnts.
8/+
• y2−3 (4−2 y
2 )= y2−12+6 y
2
¿7 y2−12
• (2 x−3 y ) (5 x2− xy+4 y
2)=5 x2− xy+4 y
2 (1)
2 x−3 y (!)
10 x3−2 x
2 y+8 x y
2
−15 x2 y+3 xy
2−12 y3
10 x3−17 x
2 y+11 x y
2−12 y3
I. +/:I /9>:+
• /rodu;t of a sum and diAeren;e
( x+ y ) ( x− y )= x
2
− y
2
xample2
2a
(¿¿ 2)2−(b)2=4a4−b
2
(2a2+b )( 2a
2−b )=¿• +uare of a binomial
x
(¿¿2±2 xy+ y2)
( x ± y )2=¿
xample2
4 a
(¿¿ 2)[¿¿2+2(4 a2)(b)+b2]=16a
4+8a2
b+b2
¿(4 a
2+b )2=¿• /rodu;t of binomials I
( x+m) ( x+n )= x2+(m+n) x+mn
xample2
(a)[¿¿2+ (1+3 )a+ (1+3 )]=a
2+4a+3
( a+1 ) ( a+3 )=¿• /rodu;t of binomials II
(mx+n ) (ox+ p )=mox2+(mp+on ) x+np
xample2
(3⋅2)a
[¿¿2+ (3 ⋅1+5 ⋅2 ) a+(5 ⋅1 )]=6 a2+13a+5(3a+5 ) (2a+1 )=¿
• :ube of a binomial
x
(¿¿3±3 x2 y+3 xy
2± y
3)( x ± y )3=¿
8ultiply (1) by ea;h term in
dd the liCe terms
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1st Balik-Aral: Academic Review Program "
xample2¿=a
3−6 a2+12a−8
(a−2 )3=¿
DO NOT DISTRIBTE EXPONENTS O/ER A SM: ( x+ y)n≠ x
n+ yn
IDI+I0 F /@08I+
• 8onomial ivisor E use x ± y
z =
x
z ±
y
z and laws o- %&onnts.
xample26 x
3 y
2+12 x2 y
3
3 x2
y2
=6 x
3 y
2
3 x2 y
2+12 x
2 y
3
3 x2 y
2 =2 x+4 y
• ong ivision
xample2 (4 x3−12 x+20) ÷ (2 x+3)=2 x
2−3 x−1+ 23
2 x+3
F:9II00a$to"#ng is the pro;ess of <nding the fa;tors of a given polynomial.
• :ommon monomial fa;tor
(ax+ay )=a ( x+ y )
xamples2 6 x3
y2
−3 x2
y+9 xy=3 xy (2 x2
y− x+3)
+1 x ¿¿
x ¿ ¿ ( x+1 )( x2−2−4 )
• iAeren;e of two suares
( x2− y2 )=( x+ y)( x− y)
xamples2 (4 a2−b
2 )=(2a)2−(b)2
¿(2a+b)(2a−b) (16m
4−n12 )=(4m
2)2−(n6)2
¿ (4 m2+n6 )( 4m2−n6 ) ¿(4m
2+n6)(2m+n
3)(2m−n3)
• /erfe;t suare trinomial
x
(¿¿2±2 xy+ y2)¿ ( x ± y )2
¿ xample2 (4 a
2+4 ab+b2 )=(2a)2+2 (2a ) ( b )+(b)2
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1st Balik-Aral: Academic Review Program 3
¿(2a+b)2
• Fa;toring by grouping E it may be possible to group terms in su;h a way that ea;h
GgroupH has a ;ommon fa;tor.
xamples2 ax+bx−ay−by=(a+b ) x−(a+b ) y
¿ (a+b )( x+ y) x y
3+2 y2− xy−2= y
2 ( xy+2 )−( xy+2 )
y
(¿¿ 2−1)( xy+2)¿¿
¿( y+1)( y−1)( xy+2)
9:I++ II
. +implify the A. polynomials by
performing the indi;ated operation(s).
1. 4 x
2
−5 x+6 x
2
−2 x!.
1
2a3
b2(2a
2+5ab−b2)
5. 3 x2n( xn+1−4 x
n+5)". (w+6)(w−6)
3.
t
t (¿¿ 2+9)(¿¿ 2−5)¿
¿
$. 35u
2v3−20u
3v2
−5u2
v
6. 16 t
4n−64 t 6n
2 t 2n
8. a6−b6
a−b?. Fa;tor the A. polynomials ;ompletely.
1. a4b
3−a3b
4+a2
b6
!. 4 s2−25r
2
5. x4n− y
6n
". 16 x2−8 x+1
3. 10a3+25a−4 a
2−10
$. (2 x−3 y)2−16
1. abx+acx−bcy−aby+bcx−acy
CHAPTER III: LINEAR AND 2ADRATIC E2ATIONS
n alg!"a#$ 3+at#on is a statement that two algebrai; expressions are eual.
xamples are2
2 x+1= x−72 x
e
5=
4
x+1 sol+t#on or "oot of the euation is a single value of the variable that maCes the
euation true. If there are many solutions to a given euation, then they are ;alled
the sol+t#on st.
I09 >I0+L#na" 3+at#ons are euations whi;h only involve polynomials of degree 1.
8/+
• 5 x−5=2 x+7
• 5 x−2 x=7+5• 3 x=12
• x=4
•
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1st Balik-Aral: Academic Review Program $
•
•
3
x+4=
2
3 x−2
•
( x+4 ) (3 x−2 ) 3
x+4=
2
3 x−2( x+4 ) (3 x−2 )
• 3 (3 x−2 )=2 ( x+4 )• 9 x−6=2 x+8• x=2
•
• >9I: >I0+
• 2+a*"at#$ 3+at#ons are euations whi;h only involve polynomials of degree
! and whi;h ;an be written in the form2 a x2+bx+c=0 . o get its roots, several
methods ;an be used.
• Fa;toring
a. +olve for x in x2+5 x=−6 .
• x2+5 x+6=0 :he;Cing2
• ( x+3 ) ( x+2 )=0 If
x=−3: (−3 )2+5 (−3 )=−6
• x+3=0 ; x+2=0 If x=−2: (−2 )2+5 (−2 )=−6
• x=−3∨ x=−2 ∴ +olution +et2
{−3,−2 }b. +olve for x in 6 x
2+19 x−7=0 .
• 6 x2+19 x−7=0 :he;Cing2
• (2 x+7 ) (3 x−1 )=0 If
x=−72
:6
(−72 )
2
+19(−72 )−
7=0
• 2 x+7=0 ;3 x−1=0 If
x=1
3:6( 13 )
2
+19 (13 )−7=0
• x=−7
2∨ x=
1
3 ∴ +olution +et2
{−7
2,1
3}
•
•
uadrati; Formula E used when the polynomial is not -a$to"a!l. For apolynomial in the form ax
2+bx+c where a , b , c
• x=−b ±√ b2−4ac
2a
•
a. +olve for x in x2+5 x=−6 .
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1st Balik-Aral: Academic Review Program 6
• x2+5 x+6=0
• a=1,b=5,c=6
•
x=
−5±√ 52−4 (1)(6)
2(1)
• x=−3∨ x=−2 (+ame as what we got from fa;toriJation)
b. +olve for x in x2+6 x−8=0 .
• a=1,b=6,c=−8 :he;Cing2
• x=−6±√ 6
2−4(1)(−8)2(1)
If x=−3±√ 17:
• x=−3± √ 17
(−3±√ 17)2+6 (−3±√ 17)−8=0
• ∴ +olution +et2
{−3±√ 17}•
• +@+8 F >I0+
• syst' o- 3+at#ons is ;omposed of two or more euations in several
variables. Kiven variables x∧ y , it has the form
• a1 x+b
1 y=c
1
• a2 x+b
2 y=c
2
•
• wherein a1
, a2
, b1, b
2, c
1∧c
2 are ;onstant real numbers. In <nding the solution
set of the system, two methods ;an be used.• +ubstitution method E transform one of the euations in terms of one variable and
substitute this to the other euation.
a. etermine the solution set of the euations 2 x+ y=3∧5 x+3 y=10.
•
• +olve the <rst euation in terms of y.
• y=3−2 x
• +ubstitute this value to the se;ond euation.
• 5 x+3 (3−2 x)=10
• − x+9=10
• x=−1
• +ubstitute this value of x in the <rst euation to get y .
• y=3−2(−1)• y=5
• ∴ +olution +et2 {−1,5 }
• limination method E multiply both euations by a nonJero real number in su;h a
way that when these euations are added, one of the variables will be
eliminated.
a. etermine the solution set of the euations 2 x+ y=3∧5 x+3 y=10.
L
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1st Balik-Aral: Academic Review Program 7
•
• o eliminate one of the variables, say y , we must multiply the <rst
euation by -5 and the se;ond euation by 1.
• −3 ⋅ (2 x+ y=3 ) →−6 x−3 y=−9
• 1⋅ (5 x+3 y=10 ) →5 x+3 y=10
• dd the resulting two euations together.
• −6 x−3 y=−9
• 5 x+3 y=10
• − x=1
• x=−1
• +ubstitute this value of x in any of the euations, say the se;ond
one, to get y .
• 5 (−1 )+3 y=10
• y=5
• ∴ +olution +et2 {−1,5 }
b. etermine the solution set of the euations3 x−2 y=13∧4 x+7 y=−2.
•
• o eliminate one of the variables, say x , we must multiply the <rst
euation by " and the se;ond euation by -5.
• 4 ⋅ (3 x−2 y=13 )→12 x−8 y=52
• −3 ⋅ (4 x+7 y=−2 )→−12 x−21 y=6
• dd the resulting two euations together.
• 12 x−8 y=52
• −12 x−21 y=6
• −29 y=58
• y=−2
• +ubstitute this value of y in any of the euations, say the se;ond
one, to get x .
• 4 x+7 (−2 )=−2
• x=3
• ∴ +olution +et2 {3,−2}
•
• *9 /9?8+
• Keneral problems
a. If a re;tangle has a length that is 5 ;m less than four times its width and
its perimeter is 14 ;m, what are its dimensionsM
•
• et w 2 width of the re;tangle
• l=4w−3 2 length of the re;tangle
• +in;e the perimeter of a re;tangle is given as / = !(width) N !(length)
and substituting the variables to the euation,
• 19=2 (w )+2(4 w−3)• 19=10w−6
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1st Balik-Aral: Academic Review Program 4
• w=2.5 ;m
• l=7 ;m
•
b. dmission ti;Cets to a motion pi;ture theater were pri;ed at O" for adults
and O5 for students. If 71& ti;Cets were sold and the total re;eipts wereO!,735, how many of ea;h type of ti;Cets were soldM
•
• et x 2 number of ti;Cets sold to adults
• y : number of ti;Cets sold to students
• +in;e there are two variables, we would need two euations. If the
total number of ti;Cets sold is 71&, writing it mathemati;ally gives
• x+ y=810
• lso, sin;e the total revenue for the ti;Cets sold was O!,735, the
se;ond euation is given as
• 4 x+3 y=2,853
• >sing elimination method to remove x , we multiply the <rst
euation by -" and the se;ond one by 1.
• −4 x−4 y=−3,240
• 4 x+3 y=2,853
• y=387
• +ubstituting this value of y to any euation, say the <rst one, x is eual
to
• x+387=810
• x=423
• ∴ 576 student ti;Cets and "!5 adult ti;Cets were sold.
•
• 8ixture problems E problems whi;h involve ;ombining solutions of diAerent
;on;entrations to obtain solution of a parti;ular ;on;entration.a. etermine how many liters of a 6P and 1!P a;id solutions should be
mixed to obtain $ liters of a 1&P a;id solution.
•
• *e are looCing for the number of liters of ea;h solution should be used
given that the <nal solution is $ liters.
• et x 2 number of liters of the 6P a;id solution
• 6− x 2 number of liters of the 1!P a;id solution
• +in;e we Cnow that a;id ;on;. x number of liters of solution = number
of liters of a;id, we ;onstru;t a table for the given solutions and the <nalmixture.
• • ;id ;on;. (P) • iters of
solution
• iters of a;id
• 6P a;id
solQn
• 6 • x • 0.07 x
• 1!P
a;id solQn
• 1! • 6− x • 0.12(6− x)
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1st Balik-Aral: Academic Review Program 1&
• Final
mixture
• 1& • $ • &.1&($)
•
• dding the values found in the last ;olumn gives us the euation
• 0.07 x+0.12 (6− x )=0.10 (6) • −0.05 x=−0.12
• x=2.4 and 6−(2.4) = 5.$
• ∴ !." of 6P a;id solution and 5.$ of the 1!P a;id solution must
be used.
•
• >niformEmotion problems - problems whi;h involve using the formula
• r ⋅t =d• where r , t ∧d are the uniform rate, time of travel and distan;e travelled,
respe;tively. In applying the formula, be $ons#stnt with the units of
measurement used.
a. ne runner tooC 5 min. "3 se;. to ;omplete a ra;e while another runnerreuired " min. to run the same ra;e. he rate of the faster runner is &."
mRse;. than the rate of the slower one. Find their rates.
•
• et r 2 rate of the slower runner (in mRse;)
• r+0.4: rate of the faster runner (in mRse;)
• +in;e the runners ran the same ra;e, they travelled eual distan;es.
pplying the formula above
• • 9ate (in
mRse;)
• ime (in se;) • istan;e (in
m)
• Faster
runner
• r+0.4 • 225 •
225(r+0.4 )• +lower
runner
• r • 240 • 240r
•
• uating the values found in the last ;olumn, we have
• 225 (r+0.4 )=240 r
• 15 r=90
• r=6 and r+0.4=6.4
• ∴ he rate of the slower runner is $ mRse; while the faster one is
$." mRse;.
•
• 9:I++ III
• . Find the solution set of the
A. euations.
1. 2 (t −5 )=3−(4+t )!. 3 (4 y+9 )=7 (2−5 y )−2 y
5. x2=8 x−15
". 8w2+10w−3=0
3. 49 x2+84 x+36=0
4. 5 y2−4 y−2=0
•
• ?. Find the solution set of the
A. systems of euations.
1. 5 x+3 y=3
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(bs;issa)
(rdinate)
uadrant I(N, N)
uadrant II(-, N)
uadrant ID(N, -)
adrant III(-, -)
1st Balik-Aral: Academic Review Program 11
• x+9 y=2
!. 3 x+4 y−4=0
• 6 x−2 y−3=0
5. 6
x +
3
y=−2
4
x+7
y=−2
• :. Find the solution(s) to the A. problems.1. he smaller of two numbers is 4 less than the larger, and their sum is 56. Find the
numbers.!. he pro<ts of a business are shared among three sto;Cholders. he <rst one
re;eives twi;e as mu;h as the se;ond and the se;ond re;eives three times as
mu;h as the third. If the pro<ts for last year were O!$,"&&, how mu;h did ea;h
sto;Cholder re;eiveM5. group of women de;ided to ;ontribute eual amounts toward obtaining a
speaCer for a booC review. If there were 1& more women, ea;h would have paid
O! less. Sowever, if there were 3 less women, ea;h would have paid O! more.
Sow many women were in the group and how mu;h was the speaCer paidM". ea worth O".1& per pound is to be mixed with tea worth O".4& per pound. Sow
many pounds of ea;h should be used to obtain !3 lb of a blend worth O"."& per
poundM6. A "a*#ato" $onta#ns 76 3+a"ts o- a wat" an* ant#-"9 sol+t#on o-
w;#$; 4<= >!y ,ol+'? #s ant#-"9. How '+$; o- t;#s sol+t#on s;o+l*
! *"a#n* an* "&la$* w#t; wat" -o" t; nw sol+t#on to ! @<=
ant#-"9$. wo airplanes, travelling in opposite dire;tions, leave an airport at the same time.
If one plane averages "7& miRhr and the other averages 3!& miRhr, how long will
it taCe before they are !&&& mi apartM1. On ;o+" a-t" a t"+$) ;as l-t on an o,"n#g;t ;a+l a 'ssng" on a
'oto"$y$l la,s -"o' t; sa' sta"t#ng &o#nt to o,"ta) t; t"+$). I-
t; 'ssng" t"a,ls at an a,"ag "at o- 46 '#;" an* o,"ta)s t;
t"+$) #n @ ;". w;at #s t; a,"ag "at o- t; t"+$)
•
•
•
•
CHAPTER I/: 7DIMENSIONAL COORDINATE SYSTEM: LINES•
• n o"*"* &a#" ( x , y ) (or $oo"*#nats) is any
;ombination of two real numbers where order is signi<;ant.
Kiven an ordered pair P in a :artesian ;oordinate plane, the
<rst number x represents the %$oo"*#nat or the a!s$#ssawhile the se;ond number y represents the y$oo"*#nat or
the o"*#nat.
•
•
•
• he x and y axes are ;alled the $oo"*#nat a%s
wherein their interse;tion is ;alled the o"#g#n O whi;h
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(&, "&&)
(1&&, $&&)
(!&&, 7&&)
(5&&, 1&&&)
("&&, 1!&&)
1st Balik-Aral: Academic Review Program 15
they are solved when y is set to & while y−¿ inter;epts have the form (0, y)sin;e they are solved when x is set to &.
• y=3 x−2
•
x
•
•
y • •
•
• iCe stated above, this euation has an
unlimited number of solutions. ?ut for linear
euations liCe this, only two diAerent ;oordinates
(but getting the x−¿ y−¿ inter;eptRs would
be easiest) to ;onstru;t its graph.
•
• y= x2−3
• x •
•
•
y
•
•
•
• For uadrati; euations, a minimum of
three diAerent ;oordinates is needed to
;onstru;t its graph.
•
• I0+
•
• Kiven an euation y=2 x+400 , it ;an be seen that for
ea;h 1&&-unit in;rease in x results to !&&-in;rease unit in
y or in lowest terms, for ea;h 1-unit in;rease in x, y
in;reases by ! units. his ;onstant ratio between the rate
of ;hange y of with respe;t to x is ;alled the slo& of the
line. etting m euals the slope, its formula is given as
• m= y
2− y
1
x2− x1
, where x2
≠ x1
.
•
• aCe note that verti;al lines have +n*n* slo& sin;e
all the ;oordinates in the line have the same x-
;oordinates while horiJontal lines have < slo& sin;e all the ;oordinates in the
line have the same y-;oordinates.
•
• Find the slope of the line through (!, 1) and ?(",6).
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1st Balik-Aral: Academic Review Program 13
•
• From the line l , we ;an dedu;e that m=2 .
+in;e they are parallel, the line we are looCing for
has also the same slope. >sing the point-slope
form,• y−(1)=2( x−1)• y=2 x−1
•
• wo distin;t non-verti;al lines, l1∧l
2 , are said to be perpendi;ular to one
another if ml1ml2=−1.
• Find the euation of the line that passes through (1, 1)
and is perpendi;ular to l : y=2 x+3 .
•
• From the line l , we ;an dedu;e that m=2 .
Kiven that they are perpendi;ular, the line we are
looCing for has the slope m=−1
2. >sing the point-
slope form,
• y−(1)=−1
2( x−1)
• y=−1
2 x+
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• . raw sCet;h of the graph of the A. euations2
1. 2 x+5 y+10=0 !. y=4− x
2
5.
". y2−9 x=0 ?. Find an euation of the line satisfying the given ;onditions.
1. he slope is " and through the points (!, -5).
!. hrough the point (-!, 5) and parallel to the line 2 x− y−2=0 .
5. hrough the point (!, ") and perpendi;ular to the line whose euation is
x−5 y+10=0 .
@. T;"o+g; t; o"#g#n an* !#s$t#ng t; angl !twn t; a%s #n t; "st
an* t;#"* 3+a*"ants.
3. :. +olve the A. problems.
1. produ;erQs total ;osts ;onsist of a manufa;turing ;ost of O!& per unit and a
<xed daily overhead. Kiven that the total ;ost of produ;ing !&& units in one day is
O",3&&, determine the <xed daily overhead.7. 0#n* t; ,al+ o- ) s+$; t;at t; l#ns w;os 3+at#ons a"
3 x+6 !y=7∧9!x+8 y=15are∥.
4.
1.
8. A. RE0ERENCES
• eithold, . (!&&!). !ollege Algebra and "rigonometr. +ingapore2 /earson
du;ation sia /te td.
=2 x−1
=2 x−1=2 x−1l : =2 x+3
l : =2 x+3
8/16/2019 Balik Aral (Algebra)
http://slidepdf.com/reader/full/balik-aral-algebra 16/17
8/16/2019 Balik Aral (Algebra)
http://slidepdf.com/reader/full/balik-aral-algebra 17/17
1st Balik-Aral: Academic Review Program 16
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