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0% found this document useful (0 votes)
604 views13 pagesRD Sharma Algebra
Uploaded by
49xygm9xsfCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
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/ 13
An algebraic expression is an expression in one or more variables having different
umber of terms. Depending on the number of terms it may be monomials, binomials,
trinomials or polynomials. Like in the case of real numbers we can also use different
mathematical operations on algebraic expression. Previously we have learnt to add
and subtract the algebraic expression. In this chapter we will learn, how to multiply
or divide the algebraic expression. We will also learn how to find the linear factors
of the algebraic expression as in the case of real numbers and how to forma linear
equation in one variable and to find its solution,
Multiplication of Algebraic Expressions
When two algebraic expressions are multiplied, the result obtained is called the
product. The expressions being multiplied are called factors or multiplicands. While
multiplying algebraic expressions first multiply numerical coefficients, then list all
the variables that occur in the terms being multiplied and add the exponents of like
variables.
> Example:
Find the product of (2x? - 5x + 4) and (x? + 7x - 8).
(A) 2x*- 9x3 — 47x? + 68x + 32 (B) 2x* + 9x3 - 47x? + 68x - 32
(C) 2x*~ 9x3 -— 47x? + 68x - 32 (D) 2x*- 9x3 — 47x? - 68x - 32
{E) None of these
Answer (B)
Explanation: (2x? - 5x + 4) (x? + 7x - 8)
= 2x°(x? + 7x — 8) —5x(x? + 7x = 8) + A(x? + 2x — 8)
= 2x! + 14x? — 16x? ~ 5x? — 35x? + 4Ox + 4x2 + 28x - 32
= 2x! + 9x? — 47x + 6Bx — 32 Ss
INTERNATIONAL OLYMPIAD OF MATHEMATICS
Division of Algebraic Expressions
The process for division of algebraic expressions is similar to the multiplicatio,
Process, the only difference is that in division process we have to divide the numericy
coefficients and subtract the exponents instead of adding. The following Points
should be remembered while dividing algebraic expressions.
®> If there are numerical coefficients in the expressions to be divided, just divide
the numerical coefficient and then divide the variables by using the laws of
exponents.
> To divide the variables just subtract the exponents of like variables.
D> Example:
(ox*y”
Gey
(A) 3xy (B) 3x (C) 3xy2 (D) 3x2
(E) None of these
Answer (C)
The simplest form of
Explanation: Su. (S)oe-4 (v4) = 3x8
Factorisation
Factorisation of an algebraic expression is the process of writing
expression as a product of two or more linear factors. Each multip|
expression is called factors of the algebraic expression. Thus the
the given algebraic expression into the product of two ormore
factorisation.
of spliting
ctors is called
Factor Theorem
According to the factor theorem if f(x) is polynomial which is completely divisible by
another polynomial g(x) = x —a,, then x —a is called the factor of the polynomial f(x)
and f(a) = 0 for all values of a.
Methods of Factorisation
|
Different algebraic expressions can be factorised by different methods. Monomials
can be easily written into their linear factors. Binomials can be factorised by using PTER 3 ALGEBRA |
" jdentities which you have learnt in previous classes. The quadratic equation can be
fed by spliting the middle term and cubic equation can be factorised by first |
it by linear factors and then reducing it to the quadratic form and then
the middle term. The other methods of factorisation are by grouping the
s having the common coefficients or having some common variables.
|
Greatest Common Divisor
Greatest common divisor or simply written as GCD of a polynomial is the largest
mo omial which is a factor of each term of the given polynomial. The common factors
of the given expression or polynomial are the GCD and the quotient thus obtained.
> Example:
y+ 4y+3
(A) (y+ 1) (¥+2) (8) (y+) (y+3)
(c) (y+2)(y +3) (0) (y+3)(y+4)
{E) None of these
Me Answer (B) 3
Explanation: y? + 4y+3=y'+3y+y+3
=yly +3) + Uy +3) =(y+ 3)y+ 1) INTERNATIONAL OLYMPIAD OF MATHEMATICS
Linear Equations in One Variable
A statement which states that two algebraic expressions are equal is called 3,
equation and an equation involving any one variable is called a linear equation in
one variable. The expression on the left of the sign of equality is called LHS and the
expression on the right is called RHS. The value of the variable which satisfies the
Biven equation is called solution of the linear equation. We can find the solution of
the linear equation either by hit and trial method or by solving the given equation
for the required variables.
Properties of an Equation
Following are some Properties of an equation:
® If same quantity is added to both sides tl
®> If same quantity is subtracted from bot!
are also equal.
® Ifboth sides of an equation are multiplied
fhe sums on both sides remains equal
th sides of an equation, the differences
by the same quantity, then the products
are equal,
> Dividing both sides of an equation by the same quantity does not alter the sit
of equality.
> Changing LHS into RHS or vice-versa des not alter the sign of equality
Transposition
We know that every equation has two sides LHS and RHS connected with the sign
equality. Sometimes both sides of the equat
contain both constants and variable
In such type of cases we transpose rarsposin cide ofthe equality and const
to another side. So the process of transposing an
‘and variables in an equatio®
is known as transposition. i f
The method of transposition involves the following steps:
Step 1: Identify the variables and constants in the eq
Step 2: Transpose the variables on LHS and constants on 0
Step 3: Simplify the equation to get the solution of the equation.
p> Example:
3x-2_2n+3_2_
Solve for x: “G—-—g—-=g-*
(a) 0 HAPTER 3 _
(c) 2 (D) -2
(E) None of these
Answer (C)
. 3x-2 2x+3_2
Explanation: We hi ——-S = =S-
pl fe have, 7 a ae
Multiplying both sides of the equation by 12, we get
=> 9x-6-8x-12=8-12x>
-18=8-12x
=>X+12x=8+ 18> 13x=26 >x=2
Practice Exercise
1. Add the following algebraic expression:
Sess
5x°-=x+=,- a ake anda eka
Spd2eog8 14g a 4 8 16
1 1 2
(A) —[135x?+62x+35 B) —[135x? +31x+42
(®) 36 | (8) aL |
: e + fr2sx? +62x+38
(c) ql27ox +62x+85 | (p) mal ]
{E) None of these
Subtract the sum of A and B from C
~32 SS ib-ine
Where Aen Par at re
Lie a 2 ere cae
B= —pqr’ +—p’qr-— T+ pqr
oe ge ee
Se
WR -ERNATIONAL. OLYMPIAD OF MATHEMATICS
INT.
eee
c= 3 par Aptat par’ -~P'rd
10 7 7 5
(a) 2 [a reanaszparo170°t]
1 r
(8) pre gir+213pqr* ~462Pqr-190P "ar ]
(c
1 ry
Fee air +203pqr +362pqr+170p‘ar |
1
(0) ios [-166p'a'r-112pq° +462pqr-171p*ar |
(E) None of these
3. Find the product of fprvex 5 + vals [pee err
B
® ag RES
45° CV a'[x? + xy ty? + 3x24 3yz]
ae
(8) Sr soy ty's303y]
b Lene
eee + 2xy ty" +2x2+3y2] 16 rate 3 Lo
(p) tgrtgs a
13,2
a Miata ‘a2
{e) None of these
. “What must be added to the following expression to make it a perfect
= 21 ze
2 (8) 1 (C),2 ee. SOUiroas
_ None of these
ctorize: -2xy +1 -y?- x" : hy a
) (1+x-y)(1-x+y) (8) (1+x+y) (1-x=) a TICS aN
INTERNATIONAL OLYMPIAD OF MATHEMATICS —__—____— i
2 +4
Gm'+13m=4 _ 12m’ +5m+"
10. Solve: Tg S am+1
(A) m=2 (8) m=-2 (Cena 1D) 3
(E) None of these
11. Which among the following is a factor of 6m* + 15m‘ + 16m? + 4m? 4 10m
—35u
(A) 3m2+4 (B) 2m?+5
(C) 2m?+5m?+2m-6 (D) 3m?+5
(E) None of these
12. If 15a* - 16a? + 9a? — Za +6 is divided by 3a — 2 then quotient and
remainder are respectively
5
(A) Sa*— 2a? + 5a and5 (B) 5a?— 2a?+ 38 and5
a 5
(c) 6a" + 2a’ Fa and6 (D) 5a?-2a?+ aand6
(E) None of these
— 75 so that the
13. What must be subtracted from 34a - 22a3- 12a‘ — 10a?
resulting polynomial is exactly divisible by —4a? + 2a? — 8a + 30?
(A) 215 (B) -215 (C) 285 (D) -285 ~
(E) None of these
14. Find the product of (p + q +r) [(p—q)? + (q-r)? + (r — p)?).
(A) P+@er (B) p?+q?+—3pqr
(C) 2(p?+q? +r?— 3pqr) (D) p*+q? +? +2 (pq+ qr+ rp)
(E) None of these
1s, If («~ p+ y—q) = 8 and (
= Xp + xq + + = 16 i
+9) (p+ gf qd + yp + yq) , then find the value of
(A) 96 (B)
64 ¢
(E) None of these a eo
a... el us Li ee A
* {g 2nd pq + gr = pr then find the value of r.
(8) 1 (C) 18 (0D) -36
(€) None of these
which one among the following statements is correct?
(A) An equation is not true for all the values of the variables used in it..
(8) An identity is true for only some certain values of the variable,
(C) Polynomial is an algebraic expression containing one or more terms with
non-zero coefficients and the variables used in it have integer exponents. INTERNATIONAL OLYMPIAD OF MATHEMATICS _ ee
a. ‘he dimensions of a rectangular box are represented as (8 + 26) (a* «4,
and (2 ~ 28) then what will be the volume of the rectangular box?
W s+ @ +e (8) at + 160 - Bab!
© #-~.e" {D) at + Bb" + 12a"?
(©) None of these
@ me) {8} me? () me2
{© None of these intel
24. Find the Vatue of tit + ts
da hole
teh ted te0
-%
aes we (cp t=2
(£) None of these
ies
2s. Divide 84 inte two parts such that (*)” ot o
Part. Find the product of both the parts.
(A) 1078 {8) 1088
(c) 2032 (0) 3076
(E) None of these
26. In a fraction, it both the numerator and denominator are decreased b
thrice, ts equal to 2. the numerator is increased by 3 and
2
is increased by 2, the fraction becomes 3° Find the fraction.
s 7
“ws ® 3 (c)
(E) None of these Pa
one
.
a See
Riga two dight EP ALGEBRA
of the digits © number is 9. if 27 is sut
sum nt number becom:
The numbel the resulta es the number eae
Ba. the digits of the original number. Find the origi: :
ee (8) 45 =
¥ 63 (0) 81
(
{e) None of these
the sum of three consecutive even numbers is 330, then find 2 th of the
greatest number.
wr (8) 84
(Q 112 (D) 140
{E) None of these ‘
ithe side of a chess board is smaller than its perimeter by 42 cm then find
the area ofthe chess board. = : : ae :
(4) 100 em? 20-5 ga ag (BLAS ral to ae
(Q) 196 cm? an — (D) asomem? —EO-lU
TERNATIONAL Ob YMPIAD OF MATHEMATICS
2.
armament wena n van arcarer ian BOO We 11s lens than 994
find 5 of the number
w) 22 (B) 432 {c) 363 (b) 333
(E) None of these
ores prtes ae to be aatvuted Ina quiz programme, The Price vie
rare vena price ts seven-sighna ofthe price vane of the fIHE Brits wt
ve rice wale of the third pent ies of the second Brie, 1 Ws
total price value of three prizes Is £18,000, then find the price value of
third price.
w rae (a) 6088
tc) ys080 (0) 46912
(E) None of these
Runs scored by Rahul in a match ‘are 38 more than the balls faced by Virat
‘The number of balls faced by Rahul is 8 less than the runs scored by virat.
tf together they have scored 144 runs and the balls faced by Rahul are 18
wane than those faced by virat. Find the number of runs scored by Rahul
we (8) 72
73 (0) 66
{€) None of these
Find the solution set of 8m + 16 2-48 and-11m +27 > -28.
W -8