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Exercise 2.6 - Chapter 2 Polynomials class 9 ncert solutions Maths PDF Download [2022-2023]


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Question 1:

Use suitable identities to find the following products:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_2a25eb99.gif  (ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_m11a4fc28.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_m4f724c0c.gif  (iv) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_m1a6a93d8.gif

(v) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_m1126e7d7.gif

Answer:

(i) By using the identity https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_4a6053e1.gif ,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_18835f88.gif

(ii) By using the identity https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_4a6053e1.gif ,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_m36eff286.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_m769442f2.gif

By using the identity https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_4a6053e1.gif ,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_m4a8a11b4.gif

(iv) By using the identity https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_83f617e.gif ,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_1ffe913b.gif

(v) By using the identity https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_83f617e.gif ,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1710/Chapter%202_html_m168620ae.gif

 

Question 2:

Evaluate the following products without multiplying directly:

(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96

Answer:

(i) 103 × 107 = (100 + 3) (100 + 7)

= (100)2 + (3 + 7) 100 + (3) (7)

[By using the identityhttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1712/Chapter%202_html_4a6053e1.gif , where

x = 100, a = 3, and b = 7]

= 10000 + 1000 + 21

= 11021

(ii) 95 × 96 = (100 − 5) (100 − 4)

= (100)2 + (− 5 − 4) 100 + (− 5) (− 4)

[By using the identityhttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1712/Chapter%202_html_4a6053e1.gif , where

x = 100, a = −5, and b = −4]

= 10000 − 900 + 20

= 9120

(iii) 104 × 96 = (100 + 4) (100 − 4)

= (100)2 − (4)2 https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1712/Chapter%202_html_7f8b80c6.gif

= 10000 − 16

= 9984

 

Question 3:

Factorise the following using appropriate identities:

(i) 9x2 + 6xy + y2

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1714/Chapter%202_html_2643c434.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1714/Chapter%202_html_m30eddee5.gif

Answer:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1714/Chapter%202_html_145a1439.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1714/Chapter%202_html_4b035260.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1714/Chapter%202_html_m7708dad9.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1714/Chapter%202_html_31564955.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1714/Chapter%202_html_m48fec950.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1714/Chapter%202_html_m77304e6e.gif

 

Question 4:

Expand each of the following, using suitable identities:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m2d88a160.gif  (ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m530aba96.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_462af0ba.gif  (iv) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_496e33cf.gif

(v) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m2bf3e4db.gif  (vi) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_732bf4b9.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_2cb53322.gif

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m423fa8ad.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m1aca6779.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_113e1c4.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m2c1a38ac.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_462af0ba.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_ma6baa81.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_77a82d32.gif

(iv) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_496e33cf.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_39978e1a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_5ece8687.gif

(v) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m2bf3e4db.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m6f9662cc.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_6d159799.gif

(vi) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_732bf4b9.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1718/Chapter%202_html_m1b805841.gif

Question 5:

Factorise:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1725/Chapter%202_html_7b906fa3.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1725/Chapter%202_html_313f33ca.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1725/Chapter%202_html_2cb53322.gif

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1725/Chapter%202_html_7b906fa3.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1725/Chapter%202_html_2eeb0815.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1725/Chapter%202_html_313f33ca.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1725/Chapter%202_html_m2b0b4d3b.gif

 

Question 6:

Write the following cubes in expanded form:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m5d3abe5a.gif  (ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m5a0da37b.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m1fb73382.gif  (iv) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_14529e7d.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m557fd93e.gif

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m4dfb09b4.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_2236d0d8.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_369f620a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_619d0bcd.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m3d0bd1f5.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m1c4e4dd3.gif

(vi) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m15ad9eaf.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1726/Chapter%202_html_m6da18086.gif

 

Question 7:

Evaluate the following using suitable identities:

(i) (99)3 (ii) (102)3 (iii) (998)3

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1728/Chapter%202_html_m557fd93e.gif

(i) (99)= (100 − 1)3

= (100)3 − (1)3 − 3(100) (1) (100 − 1)

= 1000000 − 1 − 300(99)

= 1000000 − 1 − 29700

= 970299

(ii) (102)3 = (100 + 2)3

= (100)3 + (2)3 + 3(100) (2) (100 + 2)

= 1000000 + 8 + 600 (102)

= 1000000 + 8 + 61200

= 1061208

(iii) (998)3= (1000 − 2)3

= (1000)3 − (2)3 − 3(1000) (2) (1000 − 2)

= 1000000000 − 8 − 6000(998)

= 1000000000 − 8 − 5988000

= 1000000000 − 5988008

= 994011992

 

Question 8:

Factorise each of the following:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_m4fc39721.gif  (ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_1694a0f7.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_3bf205c2.gif  (iv) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_2bd2d10a.gif

(v) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_m457e4514.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_4ab4c877.gif

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_m4fc39721.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_m47e9a093.gif

 

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_1694a0f7.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_m2df8ff2c.gif

 

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_3bf205c2.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_m305f3ced.gif

(iv) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_2bd2d10a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_m711354ce.gif

(v) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_m457e4514.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1730/Chapter%202_html_54fddca9.gif

 

Question 9:

Verify:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1733/Chapter%202_html_m32edda55.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1733/Chapter%202_html_m14492369.gif

Answer:

(i) It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1733/Chapter%202_html_m5ac06a72.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1733/Chapter%202_html_51499c8f.gif

(ii) It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1733/Chapter%202_html_mf2aeef4.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1733/Chapter%202_html_m3e972073.gif

 

Question 10:

Factorise each of the following:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1736/Chapter%202_html_m1bc14e5a.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1736/Chapter%202_html_311a3bfc.gif

[Hint: See Question 9.]

Answer:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1736/Chapter%202_html_m1bc14e5a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1736/Chapter%202_html_m3f3b1c09.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1736/Chapter%202_html_311a3bfc.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1736/Chapter%202_html_64726ab4.gif

 

Question 11:

Factorise: https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1738/Chapter%202_html_m6a8241d2.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1738/Chapter%202_html_m5db008bd.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1738/Chapter%202_html_m7ec18bd3.gif

 

Question 12:

Verify that https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1739/Chapter%202_html_m21e196f9.gif

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1739/Chapter%202_html_m5db008bd.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1739/Chapter%202_html_m47276b45.gif

 

Question 13:

If x + y + z = 0, show that

x3 + y3 + z3 = 3xyz.

Answer:

It is known that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1743/Chapter%202_html_m5db008bd.gif

Put x + y + z = 0,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1743/Chapter%202_html_677920d3.gif

 

Question 14:

Without actually calculating the cubes, find the value of each of the following:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1745/Chapter%202_html_23f44120.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1745/Chapter%202_html_426a6364.gif

Answer:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1745/Chapter%202_html_23f44120.gif

Let x = −12, y = 7, and z = 5

It can be observed that,

x + y + z = − 12 + 7 + 5 = 0

It is known that if x + y + z = 0, then

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1745/Chapter%202_html_45d7b03e.gif

∴ https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1745/Chapter%202_html_m23a87c11.gif

= −1260

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1745/Chapter%202_html_426a6364.gif

Let x = 28, y = −15, and z = −13

It can be observed that,

x + y + z = 28 + (−15) + (−13) = 28 − 28 = 0

It is known that if x + y + z = 0, then

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1745/Chapter%202_html_45d7b03e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1745/Chapter%202_html_69b7e7f4.gif

 

Question 15:

Give possible expressions for the length and breadth of each of thefollowing rectangles, in which their areas are given:

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1748/Chapter%202_html_m35ef0e57.gif

Answer:

Area = Length × Breadth

The expression given for the area of the rectangle has to be factorised. One of its factors will be its length and the other will be its breadth.

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1748/Chapter%202_html_m721215d8.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1748/Chapter%202_html_7aeebc3.gif

Therefore, possible length = 5a − 3

And, possible breadth = 5a − 4

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1748/Chapter%202_html_1a5785e0.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1748/Chapter%202_html_147b8447.gif

Therefore, possible length = 5y + 4

And, possible breadth = 7y − 3

 

Question 16:

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1755/Chapter%202_html_m5aa7051.gif

Answer:

Volume of cuboid = Length × Breadth × Height

The expression given for the volume of the cuboid has to be factorised. One of its factors will be its length, one will be its breadth, and one will be its height.

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1755/Chapter%202_html_mecdf195.gif

One of the possible solutions is as follows.

Length = 3, Breadth = x, Height = x − 4

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1755/Chapter%202_html_m72c8742e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/99/1755/Chapter%202_html_4344da82.gif

One of the possible solutions is as follows.

Length = 4k, Breadth = 3y + 5, Height = y − 1

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