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Exercise 2.3 - Chapter 2 Polynomials class 10 ncert solutions Maths - SaraNextGen [2024]


Question 1:

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

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_43dbfdc.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_3eed57e0.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_530c5199.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_m1b21800e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_2f938eb4.gif

Quotient = x − 3

Remainder = 7x − 9

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_128fbb6e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_599c58fc.gif

Quotient = x2 + x − 3

Remainder = 8

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_m7f71824b.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1934/Chapter%202_html_26d6c596.gif

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:

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_8621b3c.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_m1ea47a54.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_m2a37bf89.gif  = https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_1857d7a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_4babacf9.gif

Since the remainder is 0,

Hence, https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_m2a37bf89.gif  is a factor of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_m3e2c44f4.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_m61b9a0cc.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_29d419f8.gif

Since the remainder is 0,

Hence, https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_62a55793.gif  is a factor of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_102ec9a.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_24b43442.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_770bf738.gif

Since the remainder https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_286abf.gif ,

Hence, https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_m741aa370.gif  is not a factor of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1943/Chapter%202_html_65da55ad.gif .

Question 3:

Obtain all other zeroes of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_m3b295b25.gif , if two of its zeroes are https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_27c9fc48.gif .

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_m44b5c2b4.gif

Since the two zeroes are https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_27c9fc48.gif ,

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_4e67fb5e.gif is a factor of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_m3b295b25.gif .

Therefore, we divide the given polynomial by https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_3c8f5a52.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_46403a45.gif

We factorize https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_m3e324550.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_4fc70e37.gif

Therefore, its zero is given by x + 1 = 0

x = −1

As it has the term https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_m14245246.gif , therefore, there will be 2 zeroes at x = −1.

Hence, the zeroes of the given polynomial arehttps://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1949/Chapter%202_html_m1ddbd9ff.gif , −1 and −1.

 

Question 4:

On dividing https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1953/Chapter%202_html_m36de13ef.gif by a polynomial g(x), the quotient and remainder were − 2 and − 2x + 4, respectively. Find g(x).

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1953/Chapter%202_html_1aa22d07.gif

g(x) = ? (Divisor)

Quotient = (x − 2)

Remainder = (− 2x + 4)

Dividend = Divisor × Quotient + Remainder

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1953/Chapter%202_html_m2a98bbf8.gif

g(x) is the quotient when we dividehttps://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1953/Chapter%202_html_m3bfbfae0.gif  byhttps://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1953/Chapter%202_html_3aa0b5ec.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1953/Chapter%202_html_3f96f9c4.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1953/Chapter%202_html_1b651793.gif

 

Question 5:

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

(i) deg p(x) = deg q(x)

(ii) deg q(x) = deg r(x)

(iii) 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.

(i) 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 https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1960/Chapter%202_html_63e06f36.gif by 2.

Here, p(x) = https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1960/Chapter%202_html_63e06f36.gif

g(x) = 2

q(x) = https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1960/Chapter%202_html_3374b580.gif  and r(x) = 0

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)

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1960/Chapter%202_html_63e06f36.gif = 2(https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1960/Chapter%202_html_3374b580.gif )

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/129/1960/Chapter%202_html_63e06f36.gif

Thus, the division algorithm is satisfied.

(ii) deg q(x) = deg r(x)

Let us assume the division of x3 + x by x2,

Here, p(x) = x3 + x

g(x) = x2

q(x) = x and r(x) = x

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

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

x3 + x = (x) × x + x

x3 + x = x3 + x

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 = (x) × x + 1

x3 + 1 = x3 + 1

Thus, the division algorithm is satisfied.

Also Read : Exercise-3.1-Chapter-3-Pair-Of-Linear-Equations-In-Two-Variables-class-10-ncert-solutions-Maths

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