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Unit 2

Polynomials

Exercise 2.1

*Question 1:

The graphs of are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

(i)
clR7 iNQdO4qnQMXOhGboixuGVE9x 5vxaeBaBzdPYYBMD8EWytADCARILUY Gp1hRlxcAwe3Iai9jHsbR3r7AGXIn9d9tqs86dLMZHDE9tbhDHGL6zG3 EfzWO11fS9mWtLF Q

(ii)
p0hQr2ywrw6QuqC3CXmLRmm8mfUgYSnKoIF9D7pp0tCl7uznxNjFaN

(iii)
gcYH1P8QLK5atXOD6mZzaa8chUqCAQHUEyvlUPJ10AuxBIQmmqyiyLx7TwFQrQXrK47jfaeTqJuAf878JN88L5n8ARIAyPruEeyx9R8S5ajX1h6fV8DoSU9M5Mk4yyA 6sZIiWs

(iv)

NJ2RbqV3mu0WJUE6m3 E4BKehxhR16lI0567qsxWA8

(v)
pgZ7G8Uofkjef6yRslb0hoMrveBa9BvM2wqM w2biMELdCefNs60NFbCIiO4XqKplT8bnaPTgPUFfGV7TsgDgwYW2ccVUslPzDRAVrjc2 iqpy PmXnCBicA y44OLON9T2JTA

(vi)
ZjjkMVEe1mzxrPoxSscF1CGO2z3n36lmdAbRIx5KrYtJfd52e2AITGHrmTCDmWO8aqFKZT 1e426

Answer:

  1. The number of zeroes is 0 as the graph does not cut the x-axis at any point.
  1. The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.
  1. The number of zeroes is 3 as the graph intersects the x-axis at 3 points.
  1. The number of zeroes is 2 as the graph intersects the x-axis at 2 points.
  1. The number of zeroes is 4 as the graph intersects the x-axis at 4 points.
  1. The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

Exercise 2.2

Question 1:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
dNimWucrpG W274u7 IzK9QNnhNS41snillyfuT9zNIQkudfSu7yF5CXPImAOGAV8cHwrIsQy7vym25E5zPtPey7t7sp6beCdGkcla0gGfdcvJpKhRLS5PTbn9nr7vDkHtqLo38eF1xbffQ5iQlpd6xYRVfcGGpTFYbqeyku1Z4aj2okgBT M yH29afo9L6OGmIzqSVz9l1iewJNnFb0PLCyHYIwivSrg6S5wNoNKjbuQPFYCn0NOzcGHZfDSnfQQmKqWi UNsIGkFaAiOQBIHm Ia7al2M0YXKt6r2hT0C9oDe5EV3HPeNx2dGc Mi bC5vamC1oe wBZR5unyDcFoGNtZame4uIzeD1oZtyaNfVbr3bNJJpCl0nEyEoj4CTPUnZztEz6fA6CJFAEb4uTnosttfdCDHDjoVievCueIEtNsntCEb9Of4HMPFqMHZJP8maBFc6RPONho9hCDWzVE5D2gI0b3fnA FqyzUV9EOtrAac6SOy2QcD88EJUJmqhVCFbDNFAJpUNFp9nEQu0dWApIHGGoiD3 rmZ0o83HJ rLbr5YiQXhn7SqK9QjB 5 RSwBXnik oVwEqGZcsMhgcs2MYXQMWw354cBYh784l lQ0I4AV2S7QuyZ7HzU1qD3UMfHnhopB59ZbrKW1HFY2n1KmJU

Answer:PnIKMCPEYAaMPgGeXsVp65QkfmILCb a8lrWA3zrgBaVIKVH S2v6dZf1VJ4b2GQPDKY6RyXa2rCpZgLXQmBWz0C0egtj92J1leZFzuyzst6kkuVgl7bvn0EFs4T2QmYcGD0UFk

The value ofis zero when i.e., when x = 4 or x = −2

Therefore, the zeroes ofare 4 and −2.

Sum of zeroes =

Product of zeroes0AopQ800FCVwL5tvYSmdPxe2XbwmuKEjXWYbhrLndq4pmGobYqVVfK QHhCRZCqJpAYz5CF7pUQzfjDUPztO9uoKq4VfayTZOtUjgBaBULNwQnZN 8 aXYiR40d6azhBVzQy66g

gzcuI6ZobA2M4jcTVLrhlaN6lRheph2gbzOFrKQXaZtCMwBP2KJX X

The value of is zero when , i.e.,

Therefore, the zeroes of 4s2 − 4s + 1 are .

Sum of zeroes =
LmGxFoVQU9tqrWYoXeZk5oMEhZ fXqdGiixDKr4Onwo3lagjgAqE8UV8fbIHA8ODXCNwCn0MXcmLn2 6NmBfT33ZNKLQN188fTXgaR219VYYaInLLRWcfBnTLbdoAyyf9I3sdyA

Product of zeroes

x9 EAWbJSMtsONADU3 nQxsJSlrrNvtFcfc8JazTh6xBahdP GHyQtF5J

The value of 6x2 − 3 − 7x is zero when

Therefore, the zeroes of

Sum of zeroes =176MD3j04b3UqTcp3ZXewYlfbYfuub04nE8dwpPY6R9wpbtwBCihoFtmFxzIS1JTzXISey q1i7ZR3Dr4yS7VFJbU7UOi0Q174fkivMiOgz50TwYIcB8H

Product of zeroes =fqe0u yP rtd0nDqVVHtkNfQgJoxmRCHCNVJZn2DY7jwE2qC7eCVK6Ayci31ZL6gNwSrwHZsHFY WR BWRc G0w4XDaFEWpMhodleeVgZ FMET78ISup6

89udaQ19gLEt2DfFWHMMOfxEJGCNIZFqnpnE7tDko6fyW 51M4zjblI 5iLAg7NQZ2dqHB2rH5F00Nj1qxQp GyCsr

The value of is zero when Therefore, the zeroes of are 0 and −2.

pHPcBlEc0L6W4ES7tjhuLAeqjPYRwL74hhq54goA3on1vblAEwunq6VwZfH jYotNbfssRHt7vTd77t4w6k3w

Sum of zeroes =

Product of zeroes =i4y1adWv6r6XWVZBpVUiyp7lkzJt9SaJoKdnKGHQSNAec i7RmZvR6WnIeAtTd0CkKotYF WyOc1J1YDiErO3t396EXw BkuqYBtWMrRIYS

5AZVbk322qgK gOzQ5Nr8EwP7JBHVkVMeUSKaIR4wGxdDR VJMjcjOp193XoJfyaFjbJw4eiSOH8DTsBUXYYhjNsVUUC2X1odopuQqcijkhQfT6mJgRQ47MXd0KVVvG14sl0dg

The value of t2 − 15 is zero whenwhenQbD0mSQBuVbzYM9xmSGbSlQkJ9YUD8lWQu3a6Vr0aqjL4jbAs Q3QaceESngM2Px03YkWDUQFtiaL3lk5NrMEnicoomxqLBnRsLAFJEIZNJzV0sftTIP3nyjvdM CHILXrw79Bw

Therefore, the zeroes of t2 − 15 are .

HNqBaHMkngzbhpLzgdBVQ8pgqUjwJow6xzzdBMfNIUWfimaJ 2cbUaVKfWXUj7 9AS PzOQc94bNBgAP4BtPVKr9ieSa6QwHGJ12 hTDcIUTXToTUrmMfrEuCJMs4I14rSjGBqw

Sum of zeroes =

L61uX8ylvdIzpTvyo7L6FJush4wybIGQ7L 1kG6FnQqLYRKN3xJcfA8tSWIdmbyK52jCAtPAGBp0sHTRUq8BeZtmJcDeXim 5H Up2KHa3T8GklXIyS1mIkYdO5zmlDwOlc9 eQ

Product of zeroes =
TU1vCYJibtfaXoQJgpwFwIL7JJmriFaEm0RQPoamvfFeJETaoOHkgkBuuN mmjMsecFismJ5SAMoOaAZn9dsS kVaPy7gl7Th8cHxXmNjc3UBeZfBmwzLt8DInEoBEq0 LGwJqM

The value of is zero when when or di C05K L6v tZFHYGEsQu1 wuqboe kkhYfFgBlQcrdr6IoFMOQUrls2nAcJl1F6HUBBwms3thA2bwLdXZyrKU3sgWVw73VhiyIHmmTHBIoYQ4wrTg4mcnn7rLTX27a6sMY6MA

Therefore, the zeroes of

Sum of zeroes =

Product of zeroeshfwsEHEryxYX1IXilKB4j0V8xtVWOa0plLH8TsmNahU85AuZnnNKVzLoIguafAO4F3NkPNF7RxuVEWGoEFUkmnP6gn kkRffhoce24G4E8X0XBt9CwgRg3maIg6Usfcc7NFORM8

*Question 2:

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

Answer:

Let the polynomial be and its zeroes be

ij0EOB8yrf4nMrp9 luEbreXDsv2geBhxs wL4hoI4ay5FGDNPzqsvsqU5ZSuRuQnTQWgJJOR9AtFK1R0DDsy05WOMcSass6Gb7Vld9Rhc5DnvTpXWg IhpgJm3K9m5vB h3ncg

Therefore, the quadratic polynomial is

(ii)

Let the polynomial be and its zeroes be

BPUmlCJHnuoDIh84rZJjplDrv2t pUWUkaoArJNTZSrYwBFpQojHX9WOpI5A nCLoXfpdtOxdMIT38RJ3tztDUfMdDY FAqqXk LDq8fhaoehHftDpTsUZS5nGA7WrQlVhL vpM

Therefore, the quadratic polynomial is

Let the polynomial be and its zeroes be

Therefore, the quadratic polynomial is

Ca V6nvfnW9di1LVOdr JNKFbzSEllAhJPUeTRIIKpkIF8diUuj8YY7i nD330lBHitMEzPSyDy3IRyEt8ChMogXkarLTLL5E CUzkodF5W1IRq7aPHwlnEC AHXhqjczu7Clbs

Let the polynomial be and its zeroes be
sVlofEu o4iKqceuoQMQl94eIkij k2AHyKTVnszC6jsdkL9crfWoPciWuARNohM9wyT4hDA gC5de7H4SXaN39ZiZvAgmXQQba2s4MnJ RQ44MZXcEHjUbN 797mR 6S0jgvb0

Therefore, the quadratic polynomial isSOERit0 4l0VfFb9NjSuO73J5lvrDL FvmrBXxWHsuW9L5MOCz0v017Q uRimPFJynhvXzxZjXcYUMhIIFuXDdplYze7Vce0N0iQ1FVez2iM6OM0 w fNe6ON897mqURmD0n 1E

VBaKH2RbysraMfRJk5pAYQsH6rN3djLULoGEgesLMvRi2U jIYOBh6lI2Ps5wJnFZQvwClJL0qffXc658eD697brd0LI2Kz 2Y1Bik5S fDTqhdbthU gnKkLRRPBw3 SbyL3 8

Let the polynomial be and its zeroes be
dugQ4HrlUQgESQn9JdPC1qf1rMRDRwyBhKd5qHahTGDJPY3Hwo4CvxGmlPWGZ51nyP2F9uhp xZYszxJwcVYkppHYtEaT4bAxYxSRmsM0Je1EG6I3xvbzCql0zz28cV Ku2DaWQ

Therefore, the quadratic polynomial iswyPkvj t0b0DIYGT

Let the polynomial beMOhbrDF9t9XnSY mD nrtckq2UjR0UMnNHW 73CH0sFPYZG3Ofkzw2qmPtvm5jHq5bpPt8BvQfbhKeP34wGtuLajJVcYpWhE1 CWFJzT5eF9lA0XFr8D8 FcxA0zPhRI 3jb N0

tHT m4eutqGQt26cHla1lkwpRbYBa6dhYKFoeX1JXJA7sJhHGSrCt3sNOisux5n8tHWM2WiRpcCCVVIRnFgVsNn7 KNndi7 6prxT7fuoE6Fkw9Qe3Hylkcxz5qpcm99Dz oGs4

Therefore, the quadratic polynomial is

Exercise 2.3

Question 1:

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:

vBHgw6dqFqjhVfHrZ1ugnsI95KUybhRYaZ2ypAIQDU9ElH0HPiH4b87dDLtu2tXvsbMVOX d9NxtFdnx myIcHbFkLk LNNenXe8W3tUYR8GUoteqJt3STcHj5hm8542lUSo8s

Answer:
qwXBakbys8fJlfSv Nih0YhSV65DSq505Nxs0WQVjISIWO8BVhHMXepVM0XVRDSpsuaBIBhwEafo

Quotient = x − 3 Remainder = 7x − 9

iXmq2GtWeg5kCkE8whk

Quotient = x2 + x − 3 Remainder = 8
FrKKyVSfEY 1HCVugJf0U4jFPB5UUModqTz2IGXWBjLIyQctj42IOYzt5NtZu71APUldHzKMsc0N65HY5JctSQbPyn23Mfq w4bHYGErOzOrEQDdNyCZc1tIIk6Ip3tyCBcDhfo

LGRmZVxwNDRM7hEqFuzYl7J3RsPi8JSNLMlV0w7 MigQz6MZ0i P mxGp5 ph3NlElIWadvQFaDBOSkEYxZjtr2MgOhijRTNtBB fKuFugwGwtSkltwCLvIh69P9736lyrfY97Y

Quotient = −x2 − 2

Remainder = −5x +10

Question 2:

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

J Chsl8H5aE3CnvonrfoZkUS64xk5nAPDcD5QESJf6z931kUra2NHOUlpseSrV4DCR E8omkSUCI5 KmW0qPm aaq O olej6pfrv0djk2G6lVEKbney4J3ByTaNjFBc7QJRsog

Answer:
ca9yX4TnlgXVVbwLpFq7aWJi0XXRQSzh7FlZvzF45xHGOhNMh27OueBSzGXgnIn7AIJh2 Y7Bwvj2Kzfhcxbu3Ai8D XFdvnVGCxNyvVsDecjPhucLuCwJEpgVRQ GwzxlP2 PsJAFDqCh8 W8SdVJJ7FT4YKRrEndpq8wVhUcmnuvLPysRYao3OQOkQhEiQGhnoqqstxUQ2f2Riu712ofyMgYgeWjfmRO2LHL7J YRJNaejXN7Q3KuyZFG2BkBUg8s1ldUq KS3RwNSmWChIQg am2Oce9 UDGStykygWvunD0OrrsPGMho2TiJ3 rG8PqebJSuA DkipHEE45n3 CTfrf3KSTc 3oy po4G6gzIEMwyKeO2qsUjKSOWYEhUaUDB8ZA NMMaX QE7vSQ

          =
f31mwX0 joR7fVZbORYpHHQbH1TJWYDpjsVPpUpcFAsQa2FpvVkIueHKd xHsYdUDM2ME5FCev6hxxu6HeKKMMXv8UgiuAWaxNRlVY9LWOBjTBT6ptiANz 9eiAxZRYqyL PQ

Since the remainder is 0,

Hence,  JAFDqCh8 W8SdVJJ7FT4YKRrEndpq8wVhUcmnuvLPysRYao3OQOkQhEiQGhnoqqstxUQ2f2Riu712ofyMgYgeWjfmRO2LHL7J YRJNaejXN7Q3KuyZFG2BkBUg8s1ldUq KS3Rw is a factor of .

vZn1gOpWwCpUPPIR8fUubW5s37TMBBQIhatAU5cv3bAI6JbZtJDYzy4j9UpVP87bU1xI6Tb90bS6SEBQRtX0BkOP YHnRT6eLAUOlyc5YgpOO7G8hiFuyLkwdW2hhCGBe3tlvVo
DqI4jXPOpiJgWNtTWspJqsI8U Qpxo1bDLTsYJl6O0t2IUzwZsPJYxMomvD0 9kNkK2lUk10Br0K9K5jatWA 4d1zsC1 nbZzENm13WqKQxtbGjmSSF2Sc6z

Since the remainder is 0,

Hence, is a factor of

KTOGGaI i7DDmbYvj rX7G51dbJ lcAHyBkXQYsxPV4CfvZbOnajQxNIjwBAelkHHbWdzuQGHhR l

Since the remainder

Hence,is not a factor of

*Question 3:

Obtain all other zeroes of if two of its zeroes are

Answer:

3FmYi lDnD VNV8 qZ3PX05Q2b 7Wu2G 60MuZaylPDfEXJpmp eJvbJIKz0VJOSWEGWF4JUkXlRaQrmbn9G

Since the two zeroes are,

is a factor of

Therefore, we divide the given polynomial by
dO5U0UC Xgp5W3PMLCsITMt7KZ aHLdYuKQSwL5UX2Ggg2uPkjDUB7NSbWPGUuLdKeLPpRnN7jPWwWJpt34gxLVPPEW8d4hR8CVOPW2lEv2Zb1m4LF8S 1L4NRrE1Jpo5ibiQKM

We factorize
H8MGK68VfVPsfIMHdaGIXd1T1qZ408yr huO1hniOBpuUFzOkQm MeONGjO23aKltt8SpFFo3xpWQMzve9I36U25shsAvbTtGbAik6J058T17tYdSnh0MnvPlwANbw Gu71VVaQ

Therefore, its zero is given by

As it has the term therefore, there will be 2 zeroes at x = −1.

Hence, the zeroes of the given polynomial areand −1.

*Question 4:

On dividingby a polynomial g(x), the quotient and remainder were x − 2 and − 2x + 4, respectively. Find g(x).

Answer:yhe7mdfZBlEU xgDI7y5PVLlvzPpGUQm1CbMwLIJkrgcOknBa9rHvA2qjKlw E9Uk

g(x) = ? (Divisor)

Quotient

Remainder

Dividend = Divisor × Quotient + Remainder
8ij1O6XZPaCQU3rL CBIZ0ODxQkZ8g1wRodRbtaISgUjWyuSq0yluMee1CjDlfcWWHQhcP7otVF3vv9qd26jPFi ZB6cN 6kQcWtabdj2oPN3 89aKwIq0CLni c8ySlK FeJXA

g(x) is the quotient when we divideby

VYB2qFbmv8rk8JCdlOEwiCHrLvEEZ5Xg2KU9Qk1GLA509t49OT9ET2bLAGfGtTI1ape5F3F5Owqe8v BmW9qIZqU3RlM kU SUMwc7lmNoiF0MF6g01Mp zR mHMLTv61Jb6jE

66SefePROCm6PhUgUlBoBQ2RNIRLyJDoNfMwiPYFDNRGVCE7mDN1CvjecVvlDuOrCHvHqYzO7zmhf47twIH

Question 5:

Give examples of polynomial p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

  1. deg p(x) = deg q(x)
  2. deg q(x) = deg r(x)
  3. deg r(x) = 0

Answer:

According to the division algorithm, if p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x)

Degree of a polynomial is the highest power of the variable in the polynomial.

  1. deg p(x) = deg q(x)

Degree of quotient will be equal to degree of dividend when divisor is constant ( i.e., when any polynomial is divided by a constant).

Let us assume the division of by 2.

Here,

g(x) = 2

wEwI2wBP5V tmdNX3i9EfN86FtGXbqIQSQ4UWlB97a9RxXjOa6z i4qlDxMqLkS1GOMTtVmMli20YgzkQTT4Dhye5TQSreaoOC3qW4ESSxbIFygxIQ kwGFbYZRqJu50ZpXXZOQ

Degree of p(x) and q(x) is the same i.e., 2. Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

Xd39cp5q 3qPejs1SdskhByzDyv7uIhJ6hBLZLiYBgGWDZqNMRFN7UzIjsouIjefodz6XtnT21OENoS lE4p8dHlhU3HwqTlNUksINfN rlvn7CE65 DpUJ06Ty9wZnT0O9sO3s

Thus, the division algorithm is satisfied.

  1. deg q(x) = deg r(x)

Let us assume the division of

Here,

h2n AdJjODJXMnfQoP MOMT1k8RMCSMlTKKsulxMn8rMvzuiKh Z92Ps37 76esEE8zReTdhBe8kcFUAoPuAkhsaJSXk

Clearly, the degree of q(x) and r(x) is the same i.e., 1. Checking for division algorithm,

LUK9HDYXIrSRlgmeJr M2pcgB9QNfmBAR7WSdXBtZ7rGu E39WAYjaGPJmoxj6 DVKiyRm7PTl2r7j3kqw0Hwf9fbKsqA2M7QbWCBNRP3bRxZo7VEhia0JPZKqw5 a fQ3sPTs8

Thus, the division algorithm is satisfied. (iii)deg r(x) = 0

Degree of remainder will be 0 when remainder comes to a constant. Let us assume the division of x3 + 1by x2.

Here, p(x) = x3 + 1 g(x) = x2

q(x) = x and r(x) = 1

Clearly, the degree of r(x) is 0. Checking for division algorithm, p(x) = g(x) × q(x) + r(x)

x3 + 1 = (x2 ) × x + 1 x3 + 1 = x3 + 1

Thus, the division algorithm is satisfied.

Exercise 2.4

Question 1:

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
epj3w7rMZFR4L1chlp CBrb3

Answer:

(i)nTyfiV3x5usWgyrtnqr 5vm9qpL8Emx9XVH7y0D7Eni7WMU1MvCCRnBO8DPASHj1q4JcTCwoNl4RFQo0 0O CXoBSt6d4tMojM

Therefore,are the zeroes of the given polynomial.

Comparing the given polynomial withwe obtain

Therefore, the relationship between the zeroes and the coefficients is verified.

(ii)7kD4In8wSXunVJmdJr6o A8fnoVjJbS8puoHbjPDIkQXoEDbEQG3BCq983fyrLVmh1Z3INnISaVqSuofnsHHHGAGwpe9486 01g1YXCG39izIyo9DQK9PRuH4YVdY8MTkTFt4 QO32su9MqYTQdoIHYoHP2y054foC0la9fqzgjWZ7hygZkRep b91famk9JOwRcHgdIdbA4WWWUANXl3w7JltZsomWG5IQLBA5uIhgjEH5SFUqB8YlCPbudF9 lvTCOQapn6rWeiw

INLcLG 80x 2hcGx5Hyijuq0lzztwJMIrTBH9O1klRWiCYPYiWQEzN6uH96pRlAA 2y3ZafC 1dx Oj DeBCKdNIq7XvYzPu09UhK3NPfgjKIaqAhi yMpHsFPtZZD1Nz WCkds

Therefore, 2, 1, 1 are the zeroes of the given polynomial.

Comparing the given polynomial with , we obtain a = 1, b = −4, c = 5, d =−2.

Verification of the relationship between zeroes and coefficient of the given polynomial

1wOmRUHHyikjpPGkuSxAE6eUugoUsY9Mf71MNEb4SKJC1gcQ5Vwymvk7AD22a hPuMPVeVZrl2

Multiplication of zeroes taking two at a time tAetwGrRC3s2iqHhRiknEz upzSMcqCh1F0uO1BrRFm0kydD022m7Djfu6ebUrN7V42cnUzvF 54 TON7t7ATsF6UF74dxJZC0Wl60m6DlGdJhJnP9 CA70OJewENDdBZszzeLQ

Multiplication of zeroes

Hence, the relationship between the zeroes and the coefficients is verified.

Question 2:

Find a cubic polynomial with the sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, − 7, − 14 respectively.

Answer:

Let the polynomial be and the zeroes bebe

It is given that

tSZDLdxzzZ0uwUGWI7ScjZZOk3l95ssLle3IMR2fbA2Zgq2I aJKaDQOp8S9K63vlUiHeOvoCXVo1wltigitUW7Da6p1UYncu6nMEh8R5gffHbhrrZNvnCjvwfUPIMcTSvjw3k

If a = 1, then b = −2, c = −7, d = 14

Hence, the polynomial is

Question 3:

If the zeroes of polynomialare find a and b.

Answer:qPDHWwnxSPtfTVqCX3OEr1CbhwEeyzcjXr9dJgYsFkeZ31oIs7pul2pbdGZ2iVLTlS4MxEuXFfNFuCCyLT7TrBmVj Vk1CF278qz

Zeroes are a − b, a + a + b

Comparing the given polynomial with we obtain

p = 1, q = −3, r = 1, t = 1
0ylKtWrn4prq eYb8v9kdQspQDUHZq1kiyDPF1yBcrSAsk Z3lIzaYWgTpPKBPNNWgn8qV Ujlf 5TULzTnK9rOyuIrK6G9LzFcrMU5dnZ VUe6wTYGz smShmPHoq ghbDgMA

The zeroes are

fZ1tb32GbtRTebuqgFCJbweEgYIobeXVb2MywLUjme TUi0EuWCfCBlV

Hence, a = 1 and or

*Question 4:

If two zeroes of the polynomial are  find other zeroes.

Answer:

Given that andare zeroes of the given polynomial.

Therefore,

= x2 − 4x + 1 is a factor of the given polynomial

Ml2qM712jtdiK1aAOTqKum5olTFDSLDZ6hh5o0z6h06yzyo2CRDOB rATwSJDFNQRE6oZiLT5bonx5 qGcqLx VUd69aY363zV3bTALVqb6XSQSr0Suz49xGUSlhL6EwKsc8J M

Clearly,

It can be observed that is also a factor of the given polynomial.

And

Therefore, the value of the polynomial is also zero when

Or x = 7 or −5

Hence, 7 and −5 are also zeroes of this polynomial.

*Question 5:

If the polynomialis divided by another polynomial         the remainder comes out to be x + a, find k and a.

Answer:

By division algorithm,

Dividend = Divisor × Quotient + Remainder

Dividend − Remainder = Divisor × Quotient
VjCCD cZoBMCSGBjArSMm5OV kV879 i8IJCrGsuO2KrmLyBBeI f8q6DdYWx 6auuOLgW4aMZ9oQVlPR dXGe6AIgvJmYafqKTvS WLRN6D tTIwp9jnkAhrmJhu0DNSd3QF8

will be perfectly divisible by

.UwnqlLjlVmFQdeoPVB6oLx5qEXsvSNzgS5gDySWM3HDtVykiumEzJQ1GUjazhgoERk3yl55Chzh9ujxPKyqpxFzr3 G1iQ4LD7sanVeuLh9g994AMsq9Cs5t koEfFcAldAbfns

Let us divideby

GmnoCih2lnywsC3fj4mY umTSRR5U7htSUtHmFgiLhXIhzIVAVivC3zZ4KZbsLU8wluXQykB

It can be observed thatwill be 0.eNPgdB2iY8 ZQAZVr1fPbw5RGBTgBBUqwa8buHIZlD8Vh N10IqXZ1n0erVfIzj54mkaGKx7uKzsQRIKbmEGhP9B4hnKFnsoWC 5XkvChwx9mQKJpEb3slh7QpSQr p1VA5Aqk

Therefore,and

For

And thus, k = 5

For

ooqJQZvfrVsr7BrMI3su4

Therefore, a = −5 Hence, k = 5 and a = −5

NCERT Solutions for Class 10 Maths Chapter 2- Polynomials

As this is one of the important topics in maths, it comes under the unit – Algebra which has a weightage of 20 marks in the class 10 maths board exams. The average number of questions asked from this chapter is usually 1. This chapter talks about the following,

  • Introduction to Polynomials
  • Geometrical Meaning of the Zeros of Polynomial
  • Relationship between Zeros and Coefficients of a Polynomial
  • Division Algorithm for Polynomials

Polynomials are introduced in Class 9 where we discussed polynomials in one variable and their degrees in the previous class and this is discussed more in detail in Class 10. The NCERT solutions for class 10 mathematics for this chapter discusses the answers for various types of questions related to polynomials and their applications. We study the division algorithm for polynomials of integers and also whether the zeroes of quadratic polynomials are related to their coefficients.

The chapter starts with the introduction of polynomials in section 2.1 followed by two very important topics in section 2.2 and 2.3

  • Geometrical Meaning of the zeroes of a Polynomial – It includes 1 question having 6 different cases.
  • Relationship between Zeroes and Coefficients of a polynomial – Explore the relationship between zeroes and coefficients of a quadratic polynomial through solutions to 2 problems in Exercise 2.2 having 6 parts in each question.

Next, it discusses the following topics which were introduced in Class 9.

  • Division Algorithm for Polynomials – In this, the solutions for 5 problems in Exercise 2.3 is given having three long questions.

Key Features of NCERT Solutions for Class 10 Maths Chapter 2- Polynomials

  • It covers the whole syllabus of Class 10 Maths.
  • After studying through these NCERT solutions prepared by our subject experts, you will be confident to score well in exams.
  • It follows NCERT guidelines which help in preparing the students accordingly.
  • It contains all the important questions from the examination point of view.

Frequently Asked Questions on NCERT Solutions for Class 10 Maths Chapter 2

Where can I get the accurate solution for NCERT Solution for Class 10 Maths Chapter 2?

At SWC’S you can get the accurate solution in PDF format for NCERT Solution for Class 10 Maths Chapter 2. The NCERT Textbook Solutions for the chapter Polynomials have been designed accurately by Mathematics experts at SWC’S. All these solutions are provided by considering the new pattern of CBSE, so that students can get thorough knowledge for their exams.

Is it necessary to solve each problem provided in the NCERT Solution for Class 10 Maths Chapter 2?

Yes. Because these questions are important from an exam perspective. These questions are solved by experts to help the students to crack exercise very easily. These solutions help students to familiarize themselves with the polynomials. Solutions are available in PDF format on SWC’S website.

List out the concepts covered in NCERT Solution for Class 10 Maths Polynomials?

The concepts are covered in NCERT Solution for Class 10 Mathematics Polynomials are introduction to polynomials, geometrical meaning of the zeros of polynomial, relationship between zeros and coefficients of a polynomial and division algorithm for polynomials.


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NCERT Solutions Class 10 Maths Chapters

  • Chapter 1 Real Numbers
  • Chapter 2 Polynomials
  • Chapter 3 Pair Of Linear Equations In Two Variables
  • Chapter 4 Quadratic Equations
  • Chapter 5 Arithmetic Progressions
  • Chapter 6 Triangles
  • Chapter 7 Coordinate Geometry
  • Chapter 8 Introduction To Trigonometry
  • Chapter 9 Some Applications Of Trigonometry
  • Chapter 10 Circles
  • Chapter 11 Constructions
  • Chapter 12 Areas Related To Circles
  • Chapter 13 Surface Areas And Volumes
  • Chapter 14 Statistics
  • Chapter 15 Probability

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