chapter 2 polynomial

8/12/2019 Chapter 2 Polynomial

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POLYNOMIAL

EXERCISE-2.1

PAGE-32

Question 1:

Which of the following expressions are polynomials in one variable and which are not? State

reasons for your answer.

(i) (ii) (iii)

(iv) (v)

Answer

Discussion

(i)

Yes, this expression is a polynomial in one variablex.

(ii)

Yes, this expression is a polynomial in one variable y.

(iii)

o. !t can be observed that the exponent of variable tin term is , which is not a wholenumber. "herefore, this expression is not a polynomial.

(iv)

o. !t can be observed that the exponent of variable yin term is #$, which is not a wholenumber. "herefore, this expression is not a polynomial.

(v)

o. !t can be observed that this expression is a polynomial in % variablesx, y, and t."herefore, it is not a polynomial in one variable.
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Question 2:

Write the coefficients of in each of the following&

(i) (ii)

(iii) (iv)

Answer

Discussion

(i)

!n the above expression, the coefficient of is $.

(ii)

!n the above expression, the coefficient of is #$.

(iii)

!n the above expression, the coefficient of is .

(iv)

!n the above expression, the coefficient of is '.

Question 3:

ive one example each of a binomial of degree %, and of a monomial of degree $''.

Answer

Discussion
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*egree of a polynomial is the highest power of the variable in the polynomial.

+inomial has two terms in it. "herefore, binomial of degree % can be written as .

onomial has only one term in it. "herefore, monomial of degree $'' can be written asx$''

Question 4:

Write the degree of each of the following polynomials&

(i) (ii)

(iii) (iv) %

Answer

Video

Discussion

*egree of a polynomial is the highest power of the variable in the polynomial.

(i)

"his is a polynomial in variablexand the highest power of variablexis %. "herefore, thedegree of this polynomial is %.

(ii)

"his is a polynomial in variable yand the highest power of variable yis -. "herefore, thedegree of this polynomial is -.

(iii)

"his is a polynomial in variable tand the highest power of variable tis $. "herefore, thedegree of this polynomial is $.

(iv) %

"his is a constant polynomial. *egree of a constant polynomial is always '.

Question 5:
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lassify the following as linear, /uadratic and cubic polynomial&

(i) (ii) (iii) (iv) (v)

(vi) (vii)

Answer

Discussion

0inear polynomial, /uadratic polynomial, and cubic polynomial has its degrees as $, -, and %respectively.

(i) is a /uadratic polynomial as its degree is -.

(ii) is a cubic polynomial as its degree is %.

(iii) is a /uadratic polynomial as its degree is -.

(iv) $ 1xis a linear polynomial as its degree is $.

(v) is a linear polynomial as its degree is $.

(vi) is a /uadratic polynomial as its degree is -.

(vii) is a cubic polynomial as its degree is %.
http://www.meritnation.com/ncert-cbse/solutions/Math/pQRlGuRXo_sla_N0L5OmqEDqyeb0eSTpGfffk3oBuweDu00_eql_#answer%23answerhttp://www.meritnation.com/ncert-cbse/solutions/Math/pQRlGuRXo_sla_N0L5OmqEDqyeb0eSTpGfffk3oBuweDu00_eql_#discussion%23discussionhttp://www.meritnation.com/ncert-cbse/solutions/Math/pQRlGuRXo_sla_N0L5OmqEDqyeb0eSTpGfffk3oBuweDu00_eql_#answer%23answerhttp://www.meritnation.com/ncert-cbse/solutions/Math/pQRlGuRXo_sla_N0L5OmqEDqyeb0eSTpGfffk3oBuweDu00_eql_#discussion%23discussion


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EXERCISE-2.2

PAGE-34-35

Question 1:

2ind the value of the polynomial at

(i)x3 ' (ii)x3 #$ (iii)x3 -

Answer

Discussion

(i)

(ii)

(iii)

Question 2:

2indp('),p($) andp(-) for each of the following polynomials&

(i)p(y) 3 y-# y1 $ (ii)p(t) 3 - 1 t 1 -t-# t%

(iii)p(x) 3x%(iv)p(x) 3 (x# $) (x1 $)

Answer
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Discussion

(i)p(y) 3 y-# y1 $

p(') 3 (')-# (') 1 $ 3 $

p($) 3 ($)-# ($) 1 $ 3 $

p(-) 3 (-)-# (-) 1 $ 3 %

(ii)p(t) 3 - 1 t 1 -t-# t%

p(') 3 - 1 ' 1 - (')-# (')% 3 -

p($) 3 - 1 ($) 1 -($)-# ($)%

3 - 1 $ 1 - # $ 3 4

p(-) 3 - 1 - 1 -(-)-# (-)%

3 - 1 - 1 5 # 5 3 4

(iii)p(x) 3x%

p(') 3 (')%3 '

p($) 3 ($)%3 $

p(-) 3 (-)%3 5

(iv)p(x) 3 (x# $) (x1 $)

p(') 3 (' # $) (' 1 $) 3 (# $) ($) 3 # $

p($) 3 ($ # $) ($ 1 $) 3 ' (-) 3 '

p(-) 3 (- # $ ) (- 1 $) 3 $(%) 3 %

Question 3:

6erify whether the following are 7eroes of the polynomial, indicated against them.

(i) (ii)

(iii)p(x) 3x-# $,x3 $, # $ (iv)p(x) 3 (x1 $) (x# -),x3 # $, -
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(v)p(x) 3x- ,x3 ' (vi)

(vii) (viii)

Answer

Discussion

(i) !f is a 7ero of given polynomialp(x) 3 %x1 $, then should be '.

"herefore, is a 7ero of the given polynomial.

(ii) !f is a 7ero of polynomialp(x) 3 x# 8 , then should be '.

"herefore, is not a 7ero of the given polynomial.

(iii) !fx3 $ andx3 #$ are 7eroes of polynomialp(x) 3x-# $, thenp($) andp(#$) shouldbe '.

9ere,p($) 3 ($)-# $ 3 ', and

p(# $) 3 (# $)-# $ 3 '

9ence,x3 $ and #$ are 7eroes of the given polynomial.

(iv) !fx3 #$ andx3 - are 7eroes of polynomialp(x) 3 (x1$) (x# -), thenp(#$) andp(-)should be '.

9ere,p(#$) 3 (# $ 1 $) (# $ # -) 3 ' (#%) 3 ', and
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p(-) 3 (- 1 $) (- # - ) 3 % (') 3 '

"herefore,x3 #$ andx 3 - are 7eroes of the given polynomial.

(v) !fx3 ' is a 7ero of polynomialp(x) 3x-, thenp(') should be 7ero.

9ere,p(') 3 (')- 3 '

9ence,x3 ' is a 7ero of the given polynomial.

(vi) !f is a 7ero of polynomialp(x) 3 lx1 m, then should be '.

9ere,

"herefore, is a 7ero of the given polynomial.

(vii) !f and are 7eroes of polynomialp(x) 3 %x-# $, then

9ence, is a 7ero of the given polynomial. 9owever, is not a 7ero of the givenpolynomial.

(viii) !f is a 7ero of polynomialp(x) 3 -x1 $, then should be '.


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"herefore, is not a 7ero of the given polynomial.

Question 4:

2ind the 7ero of the polynomial in each of the following cases&

(i)p(x) 3x1 (ii)p(x) 3x# (iii)p(x) 3 -x1

(iv)p(x) 3 %x# - (v)p(x) 3 %x(vi)p(x) 3 ax, a : '

(vii)p(x) 3 cx1 d, c : ', c, are real numbers.

Answer

Discussion

;ero of a polynomial is that value of the variable at which the value of the polynomial isobtained as '.

(i)p(x) 3x1

p(x) 3 '

x1 3 '

x3 #

"herefore, forx3 #, the value of the polynomial is ' and hence,x3 # is a 7ero of thegiven polynomial.

(ii)p(x) 3x#

p(x) 3 '

x# 3 '

x3

"herefore, forx3 , the value of the polynomial is' and hence,x3 is a 7ero of the given

polynomial.

(iii)p(x) 3 -x1

p(x) 3 '

-x1 3 '

-x3 #
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"herefore, for , the value of the polynomial is ' and hence, is a 7ero of the

given polynomial.

(iv)p(x) 3 %x# -

p(x) 3 '

%x# - 3 '

"herefore, for , the value of the polynomial is ' and hence, is a 7ero of the givenpolynomial.

(v)p(x) 3 %x

p(x) 3 '

%x3 '

x3 '

"herefore, forx3 ', the value of the polynomial is ' and hence,x3 ' is a 7ero of the givenpolynomial.

(vi)p(x) 3 ax

p(x) 3 '

ax3 '

x3 '

"herefore, forx 3 ', the value of the polynomial is ' and hence,x3 ' is a 7ero of the givenpolynomial.

(vii)p(x) 3 cx1 d

p(x) 3 '

cx+ d3 '


11/40

"herefore, for , the value of the polynomial is ' and hence, is a 7ero of the

given polynomial.

EXERCISE-2.3

PAGE-40

Question 1:

2ind the remainder whenx%1 %x-1 %x1 $ is divided by

(i)x1 $ (ii) (iii)x

(iv)x1 8 (v) 1 -x

Answer

Discussion

(i)x1 $

+y long division,
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"herefore, the remainder is '.

(ii)

+y long division,

"herefore, the remainder is .

(iii)x

+y long division,


13/40

"herefore, the remainder is $.

(iv)x1 8

+y long division,

"herefore, the remainder is

(v) 1 -x

+y long division,


14/40

"herefore, the remainder is

Question 2:

2ind the remainder whenx%# ax-1


15/40

"herefore, whenx%# ax-1 1 %xis a factor of %x%1 >x.

Answer

Discussion

0et us divide (%x%1 >x) by (> 1 %x). !f the remainder obtained is ', then > 1 %x will be afactor of %x%1 >x.

+y long division,
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s the remainder is not 7ero, therefore, > 1 %xis not a factor of %x%1 >x.

EXERCISE-2.4

PAGE-43-44

Question 1:

*etermine which of the following polynomials has (x1 $) a factor&

(i)x%1x-1x1 $ (ii)x41x%1x-1x1 $

(iii)x41 %x%1 %x-1x1 $ (iv)

Answer

Discussion
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(i) !f (x1 $) is a factor ofp(x) 3x%1x-1x1 $, thenp(#$) must be 7ero, otherwise (x1 $)is not a factor ofp(x).

p(x) 3x%1x-1x1 $

p(#$) 3 (#$)%1 (#$)-1 (#$) 1 $

3 # $ 1 $ # $ # $ 3 '

9ence,x1 $ is a factor of this polynomial.

(ii) !f (x1 $) is a factor ofp(x) 3x41x%1x-1x1 $, thenp(#$) must be 7ero, otherwise(x1 $) is not a factor ofp(x).

p(x) 3x41x%1x-1x1 $

p(#$) 3 (#$)41 (#$)%1 (#$)-1 (#$) 1 $

3 $ # $ 1 $ #$ 1 $ 3 $

sp(# $) : ',

"herefore,x1 $ is not a factor of this polynomial.

(iii) !f (x1 $) is a factor of polynomialp(x) 3x41 %x%1 %x-1x1 $, thenp(#$) must be ',otherwise (x1 $) is not a factor of this polynomial.

p(#$) 3 (#$)41 %(#$)%1 %(#$)-1 (#$) 1 $

3 $ # % 1 % # $ 1 $ 3 $

sp(#$) : ',

"herefore,x1 $ is not a factor of this polynomial.

(iv) !f(x1 $) is a factor of polynomialp(x) 3 , thenp(#$) must be', otherwise (x1 $) is not a factor of this polynomial.

sp(#$) : ',

"herefore, (x1 $) is not a factor of this polynomial.


18/40

Question 2:

@se the 2actor "heorem to determine whether g(x) is a factor ofp(x) in each of the following

cases&

(i)p(x) 3 -x%1x-# -x# $, g(x) 3x1 $

(ii)p(x) 3x%1 %x-1 %x1 $, g(x) 3x1 -

(iii)p(x) 3x%# 4x-1x1


19/40

3 -> # %< 1 A 3 '

9ence, g(x) 3x# % is a factor of the given polynomial.

Question 3:

2ind the value of k, ifx# $ is a factor ofp(x) in each of the following cases&

(i)p(x) 3x-1x1 k (ii)

(iii) (iv)p(x) 3 kx-# %x1 k

Answer

Video Discussion

!fx# $ is a factor of polynomialp(x), thenp($) must be '.

(i)p(x) 3x-1x1 k

p($) 3 '

($)-1 $ 1 k3 '

- 1 k3 '

k3 #-

"herefore, the value ofk is #-.

(ii)

p($) 3 '

(iii)
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p($) 3 '

(iv)p(x) 3 kx-# %x1 k

p($) 3 '

k($)-# %($) 1 k3 '

k# % 1 k3 '

-k # % 3 '

"herefore, the value ofk is .

Question 4:

2actorise&

(i) $-x-# >x1 $ (ii) -x-1 >x1 %

(iii) x1 $

We can find two numbers such thatpq3 $- B $ 3 $- andp 1 q3 #>. "hey arep3 #4 and q3 #%.

9ere, $-x-# >x1 $ 3 $-x-# 4x # %x1 $
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3 4x (%x # $) # $ (%x # $)

3 (%x # $) (4x # $)

(ii) -x-1 >x1 %

We can find two numbers such thatpq3 - B % 3 < andp 1 q3 >.

"hey arep3 < and q 3 $.

9ere, -x-1 >x1 % 3 -x-1


22/40

(i)x%# -x-#x1 - (ii)x%1 %x-#Ax #

(iii)x%1 $%x-1 %-x1 -' (iv) -y%1 y-# -y# $

Answer

Discussion

(i) 0etp(x) 3x%# -x-#x1 -

ll the factors of - have to be considered. "hese are C $, C -.

+y trial method,

p(-) 3 (-)%# -(-)-# - 1 -

3 5 # 5 # - 1 - 3 '

"herefore, (x# -) is factor of polynomialp(x).

0et us find the /uotient on dividingx%# -x-#x1 - byx# -.

+y long division,

!t is =nown that,

*ividend 3 *ivisor B Duotient 1 Eemainder

x%# -x-#x1 - 3 (x1 $) (x-# %x1 -) 1 '

3 (x1 $) Fx-# -x#x1 -G
http://www.meritnation.com/ncert-cbse/solutions/Math/b7oVrTF4NgJ26MLrzNQ84K9rSbe_sla_FX7enWSbxgtEAow_eql_#answer%23answerhttp://www.meritnation.com/ncert-cbse/solutions/Math/b7oVrTF4NgJ26MLrzNQ84K9rSbe_sla_FX7enWSbxgtEAow_eql_#discussion%23discussionhttp://www.meritnation.com/ncert-cbse/solutions/Math/b7oVrTF4NgJ26MLrzNQ84K9rSbe_sla_FX7enWSbxgtEAow_eql_#answer%23answerhttp://www.meritnation.com/ncert-cbse/solutions/Math/b7oVrTF4NgJ26MLrzNQ84K9rSbe_sla_FX7enWSbxgtEAow_eql_#discussion%23discussion


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3 (x1 $) Fx(x# -) # $ (x# -)G

3 (x1 $) (x# $) (x# -)

3 (x# -) (x# $) (x1 $)

(ii) 0etp(x) 3x%# %x-# Ax #

ll the factors of have to be considered. "hese are C$, C .

+y trial method,

p(#$) 3 (#$)%# %(#$)-# A(#$) #

3 # $ # % 1 A # 3 '

"herefore,x1 $ is a factor of this polynomial.

0et us find the /uotient on dividingx%

1 %x-

# Ax # byx1 $.

+y long division,

!t is =nown that,


x%# %x-# Ax # 3 (x 1 $) (x-# 4x# ) 1 '

3 (x 1 $) (x-# x1x# )

3 (x1 $) F(x (x# ) 1$ (x# )G

3 (x1 $) (x# ) (x1 $)


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3 (x# ) (x1 $) (x1 $)

(iii) 0etp(x) 3x%1 $%x-1 %-x1 -'

ll the factors of -' have to be considered. Some of them are C$,

C -, C 4, C HH

+y trial method,

p(#$) 3 (#$)%1 $%(#$)-1 %-(#$) 1 -'

3 # $ 1$% # %- 1 -'

3 %% # %% 3 '

sp(#$) is 7ero, therefore,x 1 $ is a factor of this polynomialp(x).

0et us find the /uotient on dividingx%

1 $%x-

1 %-x1 -' by (x1 $).

+y long division,

!t is =nown that,


x%1 $%x-1 %-x1 -' 3 (x 1 $) (x-1 $-x1 -') 1 '

3 (x 1 $) (x-1 $'x1 -x 1 -')

3 (x1 $) Fx(x 1 $') 1 - (x 1 $')G

3 (x1 $) (x 1 $') (x 1 -)


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3 (x1 $) (x1 -) (x1 $')

(iv) 0etp(y) 3 -y%1 y-# -y# $

+y trial method,

p($) 3 - ( $)%1 ($)-# -( $) # $

3 - 1 $ # - # $3 '

"herefore, y# $ is a factor of this polynomial.

0et us find the /uotient on dividing -y%1 y-# -y# $ by y# $.

p(y) 3 -y%1 y-# -y# $

3 (y # $) (-y-1%y 1 $)

3 (y # $) (-y-1-y 1 y1$)

3 (y # $) F-y (y 1 $) 1 $ (y 1 $)G

3 (y # $) (y 1 $) (-y 1 $)

EXERCISE-2.5

PAGES-48-49-50

Question 1:


26/40

@se suitable identities to find the following products&

(i) (ii)

(iii) (iv)

(v)

Answer

Discussion

(i) +y using the identity ,

(ii) +y using the identity ,

(iii)

+y using the identity ,

(iv) +y using the identity ,
http://www.meritnation.com/ncert-cbse/solutions/Math/Ze0Sh3eap72CMXIYWBEAjhW6dOBP6d0ioWS64eeitzk_eql_#answer%23answerhttp://www.meritnation.com/ncert-cbse/solutions/Math/Ze0Sh3eap72CMXIYWBEAjhW6dOBP6d0ioWS64eeitzk_eql_#discussion%23discussionhttp://www.meritnation.com/ncert-cbse/solutions/Math/Ze0Sh3eap72CMXIYWBEAjhW6dOBP6d0ioWS64eeitzk_eql_#answer%23answerhttp://www.meritnation.com/ncert-cbse/solutions/Math/Ze0Sh3eap72CMXIYWBEAjhW6dOBP6d0ioWS64eeitzk_eql_#discussion%23discussion


27/40

(v) +y using the identity ,

Question 2:

Ivaluate the following products without multiplying directly&

(i) $'% B $'> (ii) A B A< (iii) $'4 B A 3 ($'' 1 %) ($'' 1 >)

3 ($'')-1 (% 1 >) $'' 1 (%) (>)

F+y using the identity , where

x3 $'', a3 %, and b3 >G

3 $'''' 1 $''' 1 -$

3 $$'-$

(ii) A B A< 3 ($'' # ) ($'' # 4)

3 ($'')-1 (# # 4) $'' 1 (# ) (# 4)

F+y using the identity , where

x3 $'', a3 #, and b3 #4G

3 $'''' # A'' 1 -'

3 A$-'
http://www.meritnation.com/ncert-cbse/solutions/Math/BlfTdScll52PAsaBdDutGe43DEng_pls_vR_sla_5_sla_ggzp_pls_DtfE_eql_#answer%23answerhttp://www.meritnation.com/ncert-cbse/solutionVideo/V-9-1-2-P48-S1-Q2-1712-1.swf/9/1/2/48http://www.meritnation.com/ncert-cbse/solutions/Math/BlfTdScll52PAsaBdDutGe43DEng_pls_vR_sla_5_sla_ggzp_pls_DtfE_eql_#discussion%23discussionhttp://www.meritnation.com/ncert-cbse/solutions/Math/BlfTdScll52PAsaBdDutGe43DEng_pls_vR_sla_5_sla_ggzp_pls_DtfE_eql_#answer%23answerhttp://www.meritnation.com/ncert-cbse/solutionVideo/V-9-1-2-P48-S1-Q2-1712-1.swf/9/1/2/48http://www.meritnation.com/ncert-cbse/solutions/Math/BlfTdScll52PAsaBdDutGe43DEng_pls_vR_sla_5_sla_ggzp_pls_DtfE_eql_#discussion%23discussion


28/40

(iii) $'4 B A< 3 ($'' 1 4) ($'' # 4)

3 ($'')-# (4)-

3 $'''' # $

chapter 2 polynomial

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