rd sharma solutions for class 8 math chapter 8 division of ......rd sharma solutions for class 8...
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RD Sharma Solutions for Class 8 Math Chapter 8 Division Of Algebraic Expressions
PAGE NO 8.11:
Question 1:
Divide 5.x3 - 15.x2 + 25.x by 5x.
ANSWER:
, r -3 x + S 5.v) 5.v' - 15.v- + 25.v
/ 5.r1
- 15.r: + 25.t- 15.x-
Question 2:
Divide 4Z3 + 6Z2 - z by - z.
ANSWER:
-8 r ' - 12r + 2
25,v25.v
0
PAGE NO 8.11:
6::
0
PAGE NO 8.11:
Question 3:
Divide 9x *y - bxy+ 12xy1 by --|x y .
ANSWER:
6.v + 4 8y
A 9jry 2 9.v:y
- 6ry +12xy:
-6.yv +1 Zyi’-’-An'+
IZn--’12rr
0
PAGE NO 8.11
Question 4:
Divide 3 + 2£ y + 15Ay by 3xy.
ANSWER:
2jrv+— * + 5
3.vyj 3.r V + 2v:y +1 S.vy ' 3,vV
2ry+I5.yv 2 ry
15*y15xy
0
PAGE NO 8.11
Question 5:
Divide a2 + 7x+ 12 by x + 4.
ANSWER:
x + i
x + aYx2 + lx + 12 ' ̂ + 4 *
3.v+ 12 3.v+ 12
(1
PAGE NO 8.11
Question 6:
Divide 4J/2 + 3y+ by 2y+ 1.
ANSWER:
2v + 22y + 1> V + 3v + |
V + 2y
0
PAGE NO 8.11:
Question 7:
Divide 3.x3 + 4a2 + 5x+ 18 by x+ 2.
ANSWER:
3.x2 - 2x + 9) 3.x' + 4.x2 t 5.x » ' 3.x’ + 6.x2
18
- 2x: + 5.x + IS- l x 2- 4.v + +
9.x + 18 9.x + 18
0
PAGE NO 8.11:
Question 8:
Divide 14a2 - 53x+ 45 by l x - 9.
ANSWER:
2.x- 5 __________7.x - 9 ] 14.x2 - 53.x + 45.x
' 14.x2 - 18.x■f__________
- 35.x2 t 45.x- 35.x2 + 45.x■ ____
0
PAGE NO 8.11:
Question 9:
Divide -21 + 71 x - 31a2 - 24a3 by 3 - 8a.
ANSWER:
3at + Sx - 7_________-8.x + 3)-24.x'-31.x2 + 71 .x -21
' -24.x' + 9.x2
-40.x2 -7 1 .x - 21 -40.x2 - 15.x
56.x -2 156.x 21
+0
PAGE NO 8.11:
Question 10:
Divide 3 / - 3y3 - 4J/2 - 4y by y2 - 2y.
ANSWER:
3 / + 3,v + 2
\r - 2y j 3V4 + V ~ 4V2 - 4y 3y* - 6y'
3 / - 4v= - Ay3y> - 6 r+ ____
I f - A y2y2 - 4v
- +0
PAGE NO 8.11:
Question 11:
Divide 2 f + 1 0 / + 6J/3 + y2 + 5y + 3 by 2 f + 1.
ANSWER:
r + Sy + 3________________
2 / + 1) 2 / + IO / + 6y'+y-' + 5y + 3 / 2ys + r
10/ + 6y' * 5y + 3 10/ + 5y
6 / + 3 6y' + 3
0
PAGE NO 8.11:
Question 12:
Divide/ - 2 / + 2 / + x+ 4 by .x2 + x+ 1.
ANSWER:
jr - 3.v + 4. r + x + 1) x4 - 2 r ' + 2v: + ,v + 4
-v4 + .r’ + x2
- 3jt5 + x2 + x + 4- 3.r3 - ix 2 - 3.v■ i i_______
4 .r + 4x + 4 4.v: + 4.v + 4
0
PAGE NO 8.11:
Question 13:
Divide m3 - 1 4m2 + 37m - 26 by m2 - 12m +13.
ANSWER:
12 m + 131 J ot' 14m2 + 37m 26 m' - 12m2 + 13m
- 2m: + 24m * 26- 2m: + 24m - 26+ - +
PAGE NO 8.11:
Question 14:
Dividea4 + a2 +1 by a2 + a + 1.
ANSWER:
r - x + 1) A* + A2 +
x*+x~1
+ x'
- x ' + 1-X> - x 2 — x+ + +
a2 + x + 1 x* + x+ 1
0
PAGE NO 8.11:
Question 15:
Divide a5 + a4 + a3 + a2 + a + 1 by a3 + 1 .
ANSWER:
a2 + ,r + I___________J(® + 1 ) + .x4 + a’ + at + a + 1
' A5 + A2
A4 + X3 + X + 1 .v4 + A
A1 + 1a’ + 1
0
PAGE NO 8.11:
Question 16:
Divide 14a3 - 5a2 + 9a - 1 by 2a - 1 and find the quotient and remainder
ANSWER:
- +2v: + 9.v - 1 2v: - x
2y- 1 ) i4.v, -5.y: + 9.y-1
IO.v - 1 lO.v-5
_________ 4 _
Quotient = 7a:2 + x + 5 Remainder = 4
PAGE NO 8.11:
Question 17:
Divide 6a3 - a2 - 10x - 3 by l x - 3 and find the quotient and remainder.
ANSWER:
3-t: + 4,t + 1
2v - 3 ) 6 .v ' - . r - lt tv -3 / (>x' - 9.x2
a\~- io.v- 3&v: - I2v
2v - 32v - 3
- +0
Quotient = 3a:2 + 4a; + 1 Remainder = 0
PAGE NO 8.11:
Question 18:
Divide bx3 + 11a2 - 3 9 x - 65 by ’\3x + 13 and find the quotient and remainder.
ANSWER:
2v - S____________3.x2 + I3.v+ 13j 6.v’ + 1 l.v2 - 39.v - 65
y 6,v' + 26.x2 + 26.V
- 15.v: - 65,v - 65-15.x~-65.v-65+ + +
0Quotient = 2x — 5 Remainder = 0
PAGE NO 8.12:
Question 19:
Divide 30a4 + 11a3 - 82a2 - 12x+48 by 3a2 + 2 x - 4 and find the quotient and remainder.
ANSWER:
Quotient = 10a:2 — 3a: — 12 Remainder = 0
I0.v: - 3x - 12_____________
i x z+ l x - 4 \ i 0 x i + U x ' - S Z x ' - - 12.V-48 ' 30.x-4 + 20.v' 40.x-
-9xs - 42x: - IZv + 48 -9x' - dx2 + I lxi ______________
- 36*-' - 24.v + 48- 36*~ 24.v + 48■____•______fl
PAGE NO 8.12:
Question 20:
Divide 9X4 - 4x2 + 4 by 3a2 - 4x+ 2 and find the quotient and remainder.
ANSWER:
3*: + 4x + 2
3.r - 4.v + 2 ) 9.x* - 4.x2 + 4' 9x* + 6.x2 - I2v!
12v' 10.x~ + 4l i t ’ - ldv1 + 8a
6lv: - 8x + 4 d x --8 x + 4
____0Quotient = 3x2 + 4x + 2 and remainder = 0.
PAGE NO 8.12:
Question 21:
Verify the division algorithm i.e. Dividend = Divisor x Quotient + Remainder, in each of the following. Also, write the quotient and remainder.
Dividend Divisor
(i) 14X2 + 13x - 15 l x - 4
(ii) 15z3-20z2+13z -12 3z- 6
(iii) 6/ - 28/ + 3y2 + 30y- 9 2 / - 6
(iv) 34jt- 22x3-12^-1 Ox2-75 3x+ 7
(v) 15/-16y3 + 9 y 2 -^ y + 6 3y- 2
(vi) 4y3 + 8y+8j^ + 7 2 / -y + 1
(vii) 6y3 + 4y* + 4^ + 7y2 + 27y+ 6 2y3 +1
ANSWER:
(i)
2v ■ 3______Z x-4 \ I4.x~+ 13.v- 15
/ 14a2 - Sx
21.v 1521a - u
-3
Quotient = 2x + 3 Remainder = —3 Divisor = 7x — 4Divisor X Quotient + Remainder = (7x — 4) (2x+ 3) — 3
= ' \4 ) ? + 2 ' \x - 8 x - ' \2 - 3 = 14x2 + '\3 x -1 5 = Dividend
Thus,Divisor x Quotient + Remainder = Dividend Hence verified.
00
3r - 6 ) l 5 r ' - 2 0 ~ + 13r -1 2 7 I5 r '-3 0 r :
I0+ + I3r - 12 10r: - 20r
33r - 12 33r - 66
_________54
Quotient = 5 z2 + -yz + 11Remainder = 54 Divisor = 3z — 6Divisor x Quotient + Remainder = (3z — 6) ( 5z2 + y z + l l ) + 54
= 15z3 + 10z2 + 33z - 30z2 - 20z - 6 6 + 54 = 15z3 - 20z2 + 132: - 12 = Dividend
Thus,Divisor x Quotient + Remainder = Dividend
Hence verified,
(iii)
3/ - 5y - j
2 / - 6) 6 / - 2 8 / + 3v2 + 30i- - 97 6 / - 1 8 /
1 0 / + 3 / + 30v 9 1 0 / + 30v
3 / - 9 3 / - 9
0
Quotient = 3 y3 — 5y + -|Remainder = 0Divisor = 2y2 — 6Divisor x Quotient + Remainder =(2y2 — 6) (3y3 — 5y + |) + 0
= 6y5 - 10y3 + 3y2 - 18y3 + 30y - 9 = 6j/5 -2 8 y3 + 3y2 + 30y - 9
= Dividend
Thus, Divisor X Quotient + Remainder = Dividend Hence verified.
(iv)
3.v + 7 ) - 1 Zx* - T lx ' - 10.v: + 34.v - 75 -12.x-4 28.v'
-4,v' + 2v: - 8.V 1 30_________
6.x1 - I0.v: + 34.V-75 6.x' + 14*2
- 2 4 / + 34.v - 75 -24.x- - 56.v4- +_________
90.v - 75 90.x+ 210
-285
Quotient = — 4 a3 + 2a2 — 8x+ 30 Remainder = — 285 Divisor = 3.x+ 7Divisor X Quotient + Remainder = (3x+ 7) (— 4a3 + 2.x2 — 8x+ 30) — 285
= - 12a4 + 6a3 - 24a2 + 90x — 28a3 + 14a2 - 56x+ 210 - 285 = - 12x4 - 22a3 - 1 0a2 + 34x — 75 = Dividend
Thus,Divisor x Quotient + Remainder = Dividend Hence verified.
(v)
5 v3- 2 / - | y
3v - 2) 15 / - I6 / + 9 / - — v + 615/ - 10/
6 / + 9/ - - y -y + 66 / + 4V2 3+ —
_____________ 6 _Quotient = 5 y3 — 2 y2 + | y Remainder = 6 Divisor = 3y — 2Divisor x Quotient + Remainder = (3y — 2) (5y3 — 2y2+ J-j/) + 6
= 15y4 - 6y3 + 5y2 - 10y3 + 4y2 - y y + 6 = 15y i - 16y3 + 9j/2 - y j / + 6 = Dividend
Thus,Divisor x Quotient + Remainder = Dividend Hence verified.
2y + 5_____________V + 1 ) 4 / + 8 / + 8y + 7
4/ - 2V2 + 2v + ' -
1 0 / + 6y + 71 0 / - 5y + 5
I ly 4 2Quotient = 2y+ 5 Remainder = 11y+2 Divisor = 2 / — y + 1Divisor X Quotient + Remainder = ( 2 / — y + 1)(2y + 5) +11y + 2
= 4 / + 1 0 / — 2 / — 5y+ 2y+ 5 + 11y+ 2 = 4 / + 8 / + 8y+7 = Dividend
Thus,Divisor x Quotient + Remainder = Dividend Hence verified.
(vi)
2 / -
(vii)
y + 2y + 2________________
2 / + 1 ) 6 / + V + 4y’ + 7 / + 27y + 6 ' 6 / + 3 r
4 / + 4 / + 4 V2 + 27y + 6V + 2v
4 / + V + 25y + 6 4 / + 2
4 / + 25 v + 4
Quotient = 3 / + 2y + 2 Remainder = Ay2 + 25y+ 4 Divisor = 2 / +1Divisor x Quotient + Remainder = ( 2 / + 1) (3yz + 2y + 2) +4yz + 25y+4
= 6y5 + 4 / + 4 y 3 + 3 / + 2y + 2 + 4 /+ 2 5 y + 4 = by5+4 / + Ay3 + i f + 27y + 6 = Dividend
Thus,Divisor x Quotient + Remainder = Dividend Hence verified.
PAGE NO 8.12:
Question 22:
Divide 1 5 / + 1 6 / + y y - 9 / - 6 by 3 y - 2. Write down the coefficients of the terms in the quotient.
ANSWER:
3y
+■25 809 * 27
2) 1 5 / + 16y'- 9 / + — y - 6 1 5 / - 1 0 /
2 6 / - 9 / + y - y + 6
, , , 52 ,26y’ - - y - y-
25 10 _ _—?+T y+625 , 50
~ r ~ ~ y
80 — i980
6
16027
_2_
____________ 27Quotient = 5 / + (2 6 /3 ) / + (25/9)y+ (80/27)
Remainder = (— 2/27)Coefficient of / = 5 Coefficient of / = (26/3)Coefficient of y= (25/9)Constant = (80/27)
PAGE NO 8.12:
Question 23:
Using division of polynomials, state whether(i) x+ 6 is a factor of x1 - x - 42(ii) 4 jf- 1 is a factor of Ax1 - 1 - 12(iii) 2y - 5 is a factor of 4y* - 10 / - 1Oj/2 + 30y - 15(iv) 3 / + 5 is a factor of 6J/5 + 1 5 / + 16J/3 + fy 2 + 10 y - 35(v) 2* + 3 is a factor of z5 - 9z(vi) 2a2 - x+ 3 is a factor of 6.x5 - V + 4.x3 - 5a2 - j r - 15
ANSWER:
(i)
x - 7x + 6 jx ‘ — x — 4 2
x ‘ + 6 jc
- 7 x -4 2- 7 x -4 2
0
Remainder is zero. Hence (jxh-6) is a factor of >?-x-4200
jc- 34 jr-l)4 jf: -B j : -1 2
4jc‘ — x
— 12jc—12 -12x+ 3
-15
As the remainder is non zero . Hence ( 4x-1) is not a factor of 4a2 -13x-12
(iii)
2_v - 5
2 r - 5 v + |
) 4,4 - 10,4- lOv2 + 30v - 15 / 4\4- 10,4
+- lOi-2 + 30,< -15- lOv1 + 25v
5r - 15
52
Y The remainder is non zero,2y — 5 is not a factor of 4y 4 — 10y3 — 10y2 + 30y — 15.
(iv)
212 / + 5 / +2y - —
3 / + 5] 6 / + 15 / + I 6 / * 4 / + lOv - 357 6/ + 10/I 5 / + 6 / + 4 / + IO v-3 51 5 / + 2 5 /
6 / - 2 I / + IO v -3 56/ + lOv
2 1 / -3 5
0Remainder is zero. Therefore, 3 / + 5 is a factor of 6y5 + 15y4 + 16y3 + 4y2 + lOy - 35.
(v)r'~ 3z
z= + 3 )^ -9 - -/ r5 + 3r'
—3Z3 - ft:-3z’ - 9 r+ +
0Remainder is zero; therefore, z2 + 3 is a factor of z5 — 9z.
(vi)
ix ' + X2 - Z x - 5_________
Zx2 - .v + 3 ) 6v* - .t4 + 4.v’ - 5.<r - x - 15 7 6.V5 - 3.V4 + 9.V1
2V* - 5.v’ - 5.v: - x - 15 lx* - x’ + 3.v:
-4.r’ - Rr: - at - 15-4.v’ + lx2 - 6.v
i_______- 10tv: + 5.v- 15- lftv: + 5.v- 15
0Remainder is zero; therefore, 2a:2 — x + 3 is a factor of 6a:5 — a:4 + 4a:3 — 5a:2 — x — 15.
PAGE NO 8.12:
Question 24:
Find the value of a, if x+ 2 is a factor of 4.x4 + 2 / - 3 / + 8x+ 5a.
ANSWER:
We have to find the value of a if (x + 2) is a factor of (4x4 + 2x3 — 3x2 + 8x + 5a).
S ubstituting x = —2 in 4x4 + 2x3 — 3x2 + 8x + 5a, we get :
4 ( - 2 ) 4 + 2 ( - 2 ) 3 - 3 ( - 2 ) 2 + 8 ( - 2 ) + 5a = 0 or, 64 - 16 - 12 - 16 + 5a = 0 or, 5a = — 20 or, a = —4
If (x + 2) is a factor of (4x4 + 2x3 — 3x2 + 8x + 5a), a = —4.
PAGE NO 8.12:
Question 25:
What must be added to V + 2.x3 - 2.x2 + x - 1 , so that the resulting polynomial is exactly divisible by x2+ 2x -3?
ANSWER:
jr + 1j r + 2.x - iYx* + 2x' - 2v: +.t - I
' x t + Z x ' - \ x -_ _ •x2 + x - 1 jr + 2v - 3
- x + 2
Thus, (x — 2) should be added to (a;4 + 2a;3 — 2a;2 + x — 1) to make the resulting polynomial exactly divisible by (a;2 + 2a: — 3).
PAGE NO 8.15:
Question 1:
Divide the first polynomial by the second in each of the following. Also, write the quotient and remainder:(i) 3x2 + 4x + 5, x - 2(ii) 1 Ojx2 - 7jt+ 8, 5jt- 3(iii) 5J/3 - 6J/2 + by- 1, 5y - 1(iv) X* - x3 + 5x, x - 1
ANSWER:
3x2+ 4 x + 5 x —2
__ 3 x (x -2 )+ 1 0 (x -2 )+ 2 5
- F » )_ (x -2 )(3 x + 1 0 )+ 2 5
“ F ^ )
Therefore,quotient = 3x + 10 and remainder = 25.
10x2—7x+ 8 5x—3
2x^5x—3^ — ^5x—3^ + —
(5x—3)
(5x—3)
Therefore,quotient = 2x — and remainder = -y.
5y3- 6 y 2+ 6 y - l 5y—1
y2(5y - 1) - y ( 5y - 1) + 1 ( S y - 1)
( 5 y - l )
( 5 y - l ) ( y 2- y + l )( 5 y - l )
Therefore,Quotient = y2 — y + 1 and remainder = 0
(iv) s4—x3+5x x-1
®3 ( x - 1 ) + 5 ( x - 1 ) +5 x l
_ (x-1)(x3+5)+5 — x-1
= (x3+ 5 )+ ^
Therefore, quotient = a;3 +5 and remainder = 5.
_ y2(y2—2 )+ 3 (y 2—2 )+ 6
“ y2 2
= (y2- 2 ) ( y 2+ 3 )+ 6y2- 2
= ( y 2 + s ) + ^
Therefore, quotient = y2 + 3 and remainder = 6.
PAGE NO 8.15:
Question 2:
Find whether the first polynomial is a factor of the second.(i) x + 1 ,2 ^ + 5x+4(ii) y -2 ,3 y 3 + 5y?+ 5 y + 2(iii) 4.x2 - 5,4.x4 + 7.x2 +15(iv) 4 - z, 3Z2 - 13z+4
(v) 2 a - 3 , 10a2 - 9 a - 5(vi) 4y + 1 ,8J/2 - 2y + 1
ANSWER:
(02x2+5x+4
x + 1
2 x (x + l)+ 3 (x + l)+ li+1
_ (x + l) (2 x + 3 )+ l
(x+1)
= ( 2 x + 3 ) + 3̂
V Remainder = 1
Therefore, ( x + 1 ) is not a factor of 2x2 + 5x + 4
(h) 3̂+y sy+2_ 3y2 (y - 2 ) + l ly ( y - 2 ) + 2 7 ( y - 2 ) + 5 6
y-2
(y—2)(3y2+ lly + 27)+ 56y-2
=(3y2+ lly + 2 7 )+ |§
V Remainder = 56(y—2) is not a factor o f 3y3 +5y2+5y-|-2.
4x4+ 2+15 4x2- 5
_ x2(4x2—5)+3(4x2—5)+30 4x2—5
(4 x2- 5 ) ( x2+ 3)+ 30
4X2—5
Remainder = 30
Therefore, ( 4x2 — 5 J is not a factor of 4x4 + 7x2 + 15
(iv) 3z2- 1 3 z+4 V ' 4-z__ 3z2—12z—z+ 4
4-z__ 3z(z-4)-l(z-4)
4-z= (z—4)(3z—1)
4-z= (4-z)(l-3z)
4-z= 1 — 3 z
•.* Remainder = 0(4—z) is a factor o f 3z2 —13z+4.
10a2—9a—5 2a-3
5a(2a-3)+3(2a^3)+42a-3
(2a-3)(5a+ 3)+ 42a-3
=(5a+3)+2̂Remainder = 4( 2a—3) is not a factor of 10a2 —9a—5.
(Vi)Sy2 —2 y + l
4 y + l
2 y ( 4 y + l ) - l ( 4 y + l ) + 24 y + l
(4 y + l)(2 y —1)+ 2 4 y + l
R em aind er = 2
is not a factor o f 8y2 —2 y + l .
PAGE NO 8.17:
Question 1:
Divide:a2 - 5x+ 6 by x - 3
ANSWER:
x2—5x+6 x —3
_ x2—3x —2x+6 x—3
_ x (x —3)—2 (x —3)
(i^3)= Q c -3 )(x -2 )
(x - 3 )
= x —2
PAGE NO 8.17:
Question 2:
Divide:a £ - ay1 by ax + ay
ANSWER:
ax2 —fty2 ax+ay
= a{x2- y 2) a(x+y)
a(x+y)(x-y) a(x+;/)
= x - y
PAGE NO 8.17:
Question 3:
Divide:y - / b y ^ - /
ANSWER:
x1- y 2
(x2- ^ 2)
_ (x2-^2) ^ - ^ ) (x2- ! , 2)
= a:2 + y 2
PAGE NO 8.17:
Question 4:
Divide:aaP- + (be + ad)x+ bd by (ax + b)
ANSWER:
acx2- (bet-ad) x+bd
(ax+b)
_ acx2 +bcx+adx+bd {ax+b)
_ cx{ax+b)+d{ax+b)
{ax+b)
_ {ax+b){cx+d){ax+b)
= cx + d
PAGE NO 8.17:
Question 5:
Divide:(a2 + 2ab + tP) - (a2 + 2ac + c2) by 2a + b + c
ANSWER:
(o2+2a6+62) — (a2+ 2acfc2)
{2a+b+c)
_ (a+6)2 — (a+c)2{2a+b+c)
_ {a+b+a+c){a+b—a —c)(2 a+b+c)
_ {2a+b+c){b—c)(2a+b+c)
PAGE NO 8.17:
Question 6:
Divide:\x2 - j x - 12 by - 4
ANSWER:
- x 2-----x —124 2
—x —42
l x ( l x - 4 y 3(-2X -4 )
- x —4
(H)(Vo
= + 3
PAGE NO 8.2:
Question 1:
Write the degree of each of the following polynomials.(i) 2.x2 + 5.x2 - 7(ii) 5.x2 - 3x+ 2 (Hi) 2jf + Jr2 - 8(iv) \y7 — 12y6 + 48y5 — 10(v) 3x3 +1(vi) 5(vii) 20x3 + 12xy - 1 0 / + 20
ANSWER:
(i) Correction : I t is 2x3 + 5x2 — 7 in stead o f 2x2 + 5x2 — 7.
The degree of the polymonial 2x3 + 5x2 — 7 is 3.(ii) The degree of the polymonial 5x2 — 35x + 2 is 2.
(iii) The degree of the polymonial 2x + x2 — 8 is 2.
(iv) The degree of the polymonial -|y7 — 12y6 + 48y5 — 10 is 7.
(v) The degree of the polymonial 3x3 + 1 is 3.
(vi) 5 is a constant polynomial and its degree is 0.
(vii) The degree of the polymonial 20x3 + 12x2y2 — 10y2 + 20 is 4.
PAGE NO 8.2:
Question 2:
Which of the following expressions are not polynomials?(i) x2 + 2x"2(ii) f̂a x + x2 — x3(iii) 3y3 - \/5y + 9(iv) ax1/2 + ax+ 9.x2 + 4(v) 3.x"2 + 2X"1 + Ax+5
ANSWER:
( i) x 2 + 2x-2 is not a polynomial because — 2 is the power of variable x is not a non negative
integer.
(ii) Y /a x + x2 — x3 is not a polynomial because i is the power of variable x is not a non
negative integer.
(iii) 3y3 - \/5y + 9 is a polynomial because the powers of variable y are non negative integers.
(iv ) ax 7 + ax + 9x2 + 4 is not a polynomial because is the power of variable x is not a non
negative integer.
(v ) 3x-2 + 2x-1 + 4x + 5 is not a polynomial because — 2 and — 1 are the powers of variable x
are not non negative integers.
PAGE NO 8.2:
Question 3:
Write each of the following polynomials in the standard form. Also, write their degree.(i) x2 + 3 + 6x+ 5X*(ii) a2 + 4 + 5a6(iii) (x3 -1)(x3 -4)(iv) (y3 - 2)(j/3+ 11)(v) (a3 - f ) (a3 + j f )(vi) (a + i ) ( a + | )
ANSWER:
(i) Standard form of the given polynomial can be expressed as :
(5x4 + x2 + 6x + 3) or (3 + 6x + x2 + 5x4)The degree of the polynomial is 4.(ii) Standard form of the given polynomial can be expressed as
(5a6 + a2 + 4) or (4 + a2 + 5a6)The degree of the polynomial is 6.(iii) (x3 - 1) (x3 - 4) = x6 - 5x3 + 4
Standard form of the given polynomial can be expressed as :(x6 - 5x3 + 4) or (4 - 5x3 + x6)The degree of the polynomial is 6.(iv) (y3 - 2) (y3 + 11) = y6 + 9y3 - 22
Standard form of the given polynomial can be expressed as :(y6 + 9y3 - 22) or ( -2 2 + 9y3 + y6)The degree of the polynomial is 6.( v ) (a3 - f ) ( a 3 + M ) = a6 + ^ a 3 - ^
Standard form of the given polynomial can be expressed as :(a6 + ^ a 3 - £ ) or ( - ± + & * > + > » )
The degree of the polynomial is 6.( v i ) ( a + f ) ( a + f ) = a 2 + f a + l
Standard form of the given polynomial can be expressed as :(a2 + i a + l ) o r (1 + f a + a2)The degree of the polynomial is 2.
PAGE NO 8.4:
Question 1:
Divide 6 ̂ j^ z 2 by 3 x*yz.
ANSWER:
6»V *2 _ 6x x x x x x x y x y x z x z _ 2a.(3-2) (2-1)^(2—1) _ 2xyzS®2 yz 3x iE X 2JxyX i2i
PAGE NO 8.4:
Question 2:
Divide 15rrPri3 by 5rrPrP.
ANSWER:
15m2n3 5m 2 n2
_ 15 x m x m x n x n x n 5 x m x m x n x n
= 3 m(a-a)n (s-2)= SmPn1 = 3 n
PAGE NO 8.4:
Question 3:
Divide 24a3 h3 by -8ab.
ANSWER:
24g3&3 —Sab
_ 24xoxaxox6xdx&—8xax6
= —Sat3- 1)* /3- 1)= —3a262
PAGE NO 8.4:
Question 4:
Divide -21 abc1 by labc.
ANSWER:
—21aftc2 7abc
_ —21xaxbxcxc7 x a x b x c
= - 3 oC1- 1)^ 1" 1)^ 2- 1)= - 3 c
PAGE NO 8.4:
Question 5:
Divide 72.xyz2 by -9xz.
ANSWER:
Tixyz2—9xz
__ 7 2 x x x y x z x z—9xa;xz
= - S x d -D y z t2- 1)
= - 8 y z
PAGE NO 8.4:
Question 6:
Divide -72a4b5<̂ by
ANSWER:
—72a4ft5c8
—9a262c3_ —72xaxflxaxflx6x6x6x6x6xcxcxcxcxcxcxcxc
—9xaxaxbx6xcxcxc
= 8a(4- 2)b(5- 2)C(8- 3)= 8a2b3c5
PAGE NO 8.4:
Question 7:
Simplify:16m3y2 4m2 y
ANSWER:
16m?y2 4m2y
__ 16 x m x m x m x y x j/4 x m x m x y
== Amy
PAGE NO 8.4:
Question 8:
Simplify:32 m2n 3p2
4 mnp
ANSWER:
4 mnp__ 3 2 x m x m x n x n x n x p x p
4 x m x n x p
= 8m n2p
PAGE NO 8.6:
Question 1:
Divide x+ 2.x2 + 3 V - x5 by 2x
ANSWER:
x + 2 x 2 +3a;4 —a;5 2a;
_ _x i_ 2a;2 _i_ 3a;4 x?_2a; 2a; 2a; 2a;
= \ + x + \ x * - \ x *
PAGE NO 8.6:
Question 2:
Divide yA - 3y 3 + \ y 2 b y 3y.
ANSWER:
: yL — h i. +3 y 3 y 3 y
= \yi - y 1 + -6i
PAGE NO 8.6:
Question 3:
Divide -4a3 + 4a2 + a by 2a.
ANSWER:
—4a3+4a2+a 2a
_ —4g3 4g2 j j _2a 2a 2a
= - 2 a ( 3 - 1> + 2 a ( 2 - 1) + i
= —2a2 + 2a +
PAGE NO 8.6:
Question 4:
Divide —a:6 + 2a;4 + 4a:3 + 2a:2 by y/2x2.
ANSWER:
—x6 +2xa+4s 3 +2X2 s/2s?
_ ~^6 + 2z4 + 4X3 _j_ 2a;2•s/2®2 v ^ 2 \/2a;2 V^c2
= i k x (6 ~ 2) + V 2 x (4- 2) + 2 v /2 x (3- 2) + V T x C2 - 2>
= -^=a;4 + y /7 .^ + 'ly /T .x + \/7.
PAGE NO 8.6:
Question 5:
Divide 5a3 - 6z2 + 7zby 2z.
ANSWER:
5z3 —6z2+7z 2z
_ 5z3 6z2 + 7z2z 2z 2z
= | z ( 3 - 1) - 3 Z P - 1) + |
= \ z 2 - l z + \
PAGE NO 8.6:
Question 6:
Divide -\/3 a4 + 2-̂ /3 a3 + 3a2 — 6a by 3a.
ANSWER:
7 3 a 4+2-y/3a3+3a2—6a 3a
_ >/3a4 _|_ 2‘V/3a3 3a2 6a3a 3a 3a 3a
= - L a ( 4 - D + . L a ( 3 - i ) + a ( 2 - i ) - 2 a/3 73
= - L a 3 + - L a 2 + a — 27 3 7 3