Polynomials
![CBSE Class 10 - Maths - Polynomials - NCERT Ex 2.2 CBSE Class 10 - Maths - Polynomials - NCERT Ex 2.2](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8TWjeaSMQ1voQQhzBnrB63LelNUrvNhJwUHIWKTyTQ57eLuA_e64vLQqgAaxfKvF0cSmkhfiwFVvwwidcQ-lWBQJ64Hcpe44M3eO2-Ba85MvyL5UVjY3v2_d-Q3s90Pbk7tSK7qJA4ziB/s200/cl10maths_polyEx2.1.jpg)
(i) x2-2x-8
(ii) 4s2-4s+1
(iii) 6x2-3-7x
(iv) 4u2+8u
(v) t2-15
(vi) 3x2 - x -4
Answer: (i) x2-2x-8 = (x - 4)(x +2)
⇒x2-2x-8 = 0 when x - 4= 0 or x + 2 = 0
∴ Zeroes of x2- 2x - 8 are 4 and -2.
Sum of zeroes =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRj5fndKeUfGxSns5_6r-5cpXVZ43veYadh1N7F9_9tfdE1ZBwtcGankKgBPsap_cWotoThBFnmxch3Kfzaxgs7EMG69Nuvc7nGCdn4P2QwUcsljk2IgGTIrM7qlaLZQUmzcOFY04PpVHr/s1600/cl10_2.2_img1.gif)
Product of zeroes =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhUaNUU3eQ9PRcrLX_M__xUCEoxCix-YJgrXaMygQVwrCCHzNxo9I75WZhAUbEHQeHsnpd6-Q0o11_3LrZrSbLboCBA4RjvB7YogOdDGnEZwUJS6gjvEE8yBgPZCDaG3GYW3ed9WM63oyf/s1600/cl10_2.2_img2.gif)
Hence relationship between zeroes and coefficients are verified.
(ii) 4s2-4s+1
= (2s -1 )2
⇒ 4s2-4s+1 = 0 when 2s -1 = 0
⇒ s = 1/2
∴ Zeroes of 4s2-4s+1 are ½ and ½.
Sum of zeroes =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiohDGwTVWYtWiH02Gr4nPfzHEvhZtffnPn-uuWw0BwzySVoY1Eqlg4t7WqqCcgFGE9-7H_8DRsELhK_wx79vLhE0hyphenhyphenWVCdAQAkNUF-p-as8gsu434v7uweGsEvdiu-31fWjw0chgbTBIs/s1600/cl10_2.2_img5.gif)
Product of zeroes =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjchSdkmntCPcqhPijx5_zfaKspzV_R66_hHhf8aE08QxZYF7AudAJMqTpjaU_dI-njPaQ9LtJMALqtBTmdZO-3WhUGXIWa9CMk870E2hJrzH8W-ZaUeWLya9e3jm7f7Ferv2D9Xjtrakmb/s1600/cl10_2.2_img6.gif)
Hence relationship between zeroes and coefficients are verified.
(iii) 6x2-3-7x
= (3x + 1)(2x - 3)
⇒ 6x2-3-7x = 0 when 3x + 1 = 0 OR 2x - 3 = 0
⇒ x = -1/3 or x = 3/2 ∴ Zeroes of 6x2-3-7x are -1/3 and 3/2
Sum of zeroes =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjU5wauzkTATJdUzPOo7-pfY5Anfe5Ns_viK1ANHkdtnNy-TU5KRxTIPGmRfKHYDRA7C6liohC_rixDabtPZtyl-xakCHRbOWxQZOW-GA3NP_WPoPoxox7nkf4TXrKe0Ge3CyewBSFaVpsN/s320/cl10_2.2_img3.gif)
Product of zeroes =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcTxBk6at9rmhNznr2ilwJ6FvKyOOC3-59XIJbIofhNfSOH_RcYnZGSN2TwT76israh7c6tjsgYfnx9tYAsgUW6M7XpBvq_jkABttT7sKjJY_7g44xBkp5mFANVNVHcFlUpsEw-jyw8FBQ/s1600/cl10_2.2_img4.gif)
Hence relationship between zeroes and coefficients are verified.
(iv) 4u2+8u
= 4u(u + 2)
⇒ 4u2+8u = 0 when 4u = 0 OR u + 2 = 0
⇒ u = 0 OR u = -2
∴ Zeroes of 4u2+8u are 0 OR -2
Sum of zeroes =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBCWSePeiG-SO7f0-12Cvrxg__c20rYr3gPtG3vnDxUJLRdqjiEQTnvar4BbNHmqcSXesQxoeV2Xa5yycfGnL_-Gyk4mmGpewawir0RyiNsDx7EZpK4ogLjapp3o6BlLkus3sdXHXs1fTv/s320/cl10_2.2_img7.gif)
Product of zeroes =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgd15I2IR5_gPtARi3G6DjYO-Dw1b6ZQ7I38LVTpkTjMoOMDHVJBqa1z40P2ztjw_KXpZgCdT21OJ65gIplC371Lrpm5n8mwTboGNchdswbQBkIy0nwNg5mO9niInQo7K3gUHU6YgM9MDv3/s320/cl10_2.2_img8.gif)
Hence relationship between zeroes and coefficients are verified.
(v) t2-15
= t2 - 0t -15 = (t - √15)(t + √15)
⇒ The value of t2-15 is 0 when t - √15 = 0 OR t + √15 = 0
⇒ t = √15 or t = -√15.
Sum of zeroes = √15 -√15 = 0
Product of zeores = (√15) ☓ (-√15) = -15
Hence relationship between zeroes and coefficients are verified.
(vi) 3x2 - x -4
= 3x2 - 3x + 4x -4 = 3x(x + 1) - 4(x + 1) = (x + 1)(3x - 4)
⇒ The value of 3x2 - x -4 is 0, when x + 1 = 0 OR 3x -4 = 0
⇒ x = -1 OR x = 4/3
Sum of zeroes = -1 + 4/3 = -1/3
Product of zeroes = -1 ☓ 4/3 = -4/3
Hence relationship between zeroes and coefficients are verified.
Q2: Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) 1/4, -1
(ii) √2, 1/3
(iii) 0, √5
(iv) 1,1
(v)-1/4, 1/4
(vi) 4,1
Answer:
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