# NCERT Solutions Class 10 for Maths Chapter 2 Polynomials

NCERT Solutions Class 10 for **Maths Chapter 2 Polynomials** : In this post, we will share with you all the detailed **NCERT Solutions of Class 10 Maths Chapter 2 Polynomials**. This will contain both in-text and back-exercise questions for Science and Social Science, and all exercise questions for Mathematics. For all school and board level examinations, doing all the NCERT Questions is a must.

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# Class 10 Maths Chapter 2- Polynomials

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## Exercise 2.1 (Page 28)

**Question 1:**

The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

**Answer 1:**

Total number of zeroes in any polynomial equation = total number of times the curve intersects x-axis.

(i) In the given graph, the number of zeroes of p(x) is 0 because the graph is parallel to x-axis does not cut it at any point.

(ii) In the given graph, the number of zeroes of p(x) is 1 because the graph intersects the x-axis at only one point.

(iii) In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at any three points.

(iv) In the given graph, the number of zeroes of p(x) is 2 because the graph intersects the x-axis at two points.

(v) In the given graph, the number of zeroes of p(x) is 4 because the graph intersects the x-axis at four points.

(vi) In the given graph, the number of zeroes of p(x) is 3 because the graph intersects the x-axis at three points.

## Exercise 2.2 (Page 33)

**Question 1:**

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x^{2}–2x –8**Answer (i):**

x^{2}– 4x+2x–8 = x(x–4)+2(x–4) = (x-4)(x+2)

Therefore, zeroes of polynomial equation x^{2}–2x–8 are (4, -2)

Sum of zeroes = 4–2 = 2 = -(-2)/1 = -(Coefficient of x)/(Coefficient of x^{2})

Product of zeroes = 4×(-2) = -8 =-(8)/1 = (Constant term)/(Coefficient of x^{2})

(ii) 4s^{2}–4s+1**Answer (ii):**

4s^{2}–2s–2s+1 = 2s(2s–1)–1(2s-1) = (2s–1)(2s–1)

Therefore, zeroes of polynomial equation 4s^{2}–4s+1 are (1/2, 1/2)

Sum of zeroes = (½)+(1/2) = 1 = -4/4 = -(Coefficient of s)/(Coefficient of s^{2})

Product of zeros = (1/2)×(1/2) = 1/4 = (Constant term)/(Coefficient of s^{2 })

(iii) 6x^{2}–3–7x**Answer (iii):**

6x^{2}–7x–3 = 6x^{2 }– 9x + 2x – 3 = 3x(2x – 3) +1(2x – 3) = (3x+1)(2x-3)

Therefore, zeroes of polynomial equation 6x^{2}–3–7x are (-1/3, 3/2)

Sum of zeroes = -(1/3)+(3/2) = (7/6) = -(Coefficient of x)/(Coefficient of x^{2})

Product of zeroes = -(1/3)×(3/2) = -(3/6) = (Constant term) /(Coefficient of x^{2 })

(iv) ** **4u^{2}+8u**Answer (iv):**

4u(u+2)

Therefore, zeroes of polynomial equation 4u^{2} + 8u are (0, -2).

Sum of zeroes = 0+(-2) = -2 = -(8/4) = = -(Coefficient of u)/(Coefficient of u^{2})

Product of zeroes = 0×-2

(v) ** **t^{2}–15**Answer (v):**

t^{2} = 15 or t = ±√15

Therefore, zeroes of polynomial equation t^{2} –15 are (√15, -√15)

Sum of zeroes =√15+(-√15) = 0= -(0/1)= -(Coefficient of t) / (Coefficient of t^{2})

Product of zeroes = √15×(-√15) = -15 = -15/1 = (Constant term) / (Coefficient of t^{2 })

(vi) 3x^{2}–x–4**Answer (vi):**

3x^{2}–4x+3x–4 = x(3x-4)+1(3x-4) = (3x – 4)(x + 1)

Therefore, zeroes of polynomial equation3x^{2} – x – 4 are (4/3, -1)

Sum of zeroes = (4/3)+(-1) = (1/3)= -(-1/3) = -(Coefficient of x) / (Coefficient of x^{2})

Product of zeroes=(4/3)×(-1) = (-4/3) = (Constant term) /(Coefficient of x^{2 })

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**Question 2:**

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

**(i)** 1/4 , -1**Answer (i):**

From the formulas of sum and product of zeroes, we know,

Sum of zeroes = α+β

Product of zeroes = α β

Sum of zeroes = α+β = 1/4

Product of zeroes = α β = -1

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x^{2}–(α+β)x +αβ = 0

x^{2}–(1/4)x +(-1) = 0

4x^{2}–x-4 = 0

**Thus,4x ^{2}–x–4 is the **quadratic polynomial.

**(ii)** √2, 1/3**Answer (ii):**

Sum of zeroes = α + β =√2

Product of zeroes = α β = 1/3

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x^{2}–(α+β)x +αβ = 0

x^{2}** –(**√2)x + (1/3) = 0

3x^{2}-3√2x+1 = 0

**Thus, 3x ^{2}-3**√2x+1

**is the**quadratic polynomial.

**(iii)** 0, √5**Answer (iii):**

Sum of zeroes = α+β = 0

Product of zeroes = α β = √5

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly

as:-

x^{2}–(α+β)x +αβ = 0

x^{2}–(0)x **+**√5**= 0**

**Thus, x ^{2}+**√5

**is the**quadratic polynomial.

**(iv)** 1, 1**Answer (iv):**

Sum of zeroes = α+β = 1

Product of zeroes = α β = 1

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x^{2}–(α+β)x +αβ = 0

x^{2}–x+1 = 0

**Thus , x ^{2}–x+1is the **quadratic polynomial.

**(v)** -1/4, 1/4**Answer (v):**

Sum of zeroes = α+β = -1/4

Product of zeroes = α β = 1/4

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x^{2}–(α+β)x +αβ = 0

x^{2}–(-1/4)x +(1/4) = 0

4x^{2}+x+1 = 0

**Thus,4x ^{2}+x+1 is the **quadratic polynomial.

**(vi) **4, 1**Answer (vi):**

Sum of zeroes = α+β =

Product of zeroes = αβ = 1

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x^{2}–(α+β)x+αβ = 0

x^{2}–4x+1 = 0

**Thus, x ^{2}–4x+1 is the **quadratic polynomial.

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## Exercise 2.3 (Page 36)

**Question 1:**

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

**(i)** p(x) = x^{3}-3x^{2}+5x–3 , g(x) = x^{2}–2**Answer (i):**

Given,

Dividend = p(x) = x^{3}-3x^{2}+5x–3

Divisor = g(x) = x^{2}– 2

**(ii)** p(x) = x^{4}-3x^{2}+4x+5 , g(x) = x^{2}+1-x**Answer (ii):**

Dividend = p(x) = x^{4 }– 3x^{2 }+ 4x +5

Divisor = g(x) = x^{2} +1-x

**(iii) **p(x) =x^{4}–5x+6, g(x) = 2–x^{2}**Anwer (iii):**

Given,

Dividend = p(x) =x^{4} – 5x + 6 = x^{4 }+0x^{2}–5x+6

Divisor = g(x) = 2–x^{2} = –x^{2}+2

**Question 2:**

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.**(i)** t^{2} – 3, 2t^{4} + 3t^{3} – 2t^{2}– 9t – 12**(ii)** x^{2} + 3x + 1, 3x^{4} + 5x^{3} – 7x^{2} + 2x + 2**(iii**) x^{2} + 3x + 1, x^{5} – 4x^{3 }+ x^{2} + 3x + 1

**Answer 2:**

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**Question 3:**

Obtain all other zeroes of 3x^{4}+6x^{3}-2x^{2}-10x-5, if two of its zeroes are √(5/3) and – √(5/3).

**Answer 3:**

Since this is a polynomial equation of degree 4, hence there will be total 4 roots.

√(5/3) and – √(5/3) are zeroes of polynomial f(x).**∴ **(x –√(5/3)) (x+√(5/3)** **= x^{2}-(5/3) = 0**(**3x^{2}−5)=0**,** is a factor of given polynomial f(x).

Now, when we will divide f(x) by (3x^{2}−5) the quotient obtained will also be a factor of f(x) and the remainder will be 0.

Therefore, 3x^{4 }+6x^{3 }−2x^{2 }−10x–5 = (3x^{2 }–5)(x^{2}+2x+1)

Now, on further factorizing (x^{2}+2x+1) we get,

x^{2}+2x+1 = x^{2}+x+x+1 = 0

x(x+1)+1(x+1) = 0

(x+1)(x+1) = 0

So, its zeroes are given by: **x= −1 **and** x = −1.**

Therefore, all four zeroes of given polynomial equation are:

**√(5/3),- √(5/3) , −1 and −1.**

Hence, is the answer.

**Question 4:**

On dividing x^{3}-3x^{2}+x+2 by a polynomial g(x), the quotient and remainder were x–2 and –2x+4, respectively. Find g(x).

**Answer 4:**

Given,

Dividend, p(x) = x^{3}-3x^{2}+x+2

Quotient = x-2

Remainder = –2x+4

We have to find the value of Divisor, g(x) =?

As we know,

Dividend = Divisor × Quotient + Remainder

∴ x^{3}-3x^{2}+x+2 = g(x)×(x-2) + (-2x+4)

x^{3}-3x^{2}+x+2-(-2x+4) = g(x)×(x-2)

Therefore, g(x) × (x-2) = x^{3}-3x^{2}+x+2

Now, for finding g(x) we will divide x^{3}-3x^{2}+x+2 with (x-2)

Therefore,** g(x) = (x ^{2}–x+1)**

**Question 5:**

Give examples of polynomials 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 5:**

According to the division algorithm, dividend p(x) and divisor g(x) are two polynomials, where g(x)≠0. Then we can find the value of quotient q(x) and remainder r(x), with the help of below given formula;

Dividend = Divisor × Quotient + Remainder

∴ p(x) = g(x)×q(x)+r(x)

Where r(x) = 0 or degree of r(x)< degree of g(x).

Now let us proof the three given cases as per division algorithm by taking examples for each.

**(i) **deg p(x) = deg q(x)

Degree of dividend is equal to degree of quotient, only when the divisor is a constant term.

Let us take an example, 3x^{2}+3x+3 is a polynomial to be divided by 3.

So, (3x^{2}+3x+3)/3 = x^{2}+x+1 = q(x)

Thus, you can see, the degree of quotient is equal to the degree of dividend.

Hence, division algorithm is satisfied here.

**(ii)** deg q(x) = deg r(x)

Let us take an example , p(x)=x^{2}+x is a polynomial to be divided by g(x)=x.

So, (x^{2}+x)/x = x+1 = q(x)

Also, remainder, r(x) = 0

Thus, you can see, the degree of quotient is equal to the degree of remainder.

Hence, division algorithm is satisfied here.

**(iii) **deg r(x) = 0

The degree of remainder is 0 only when the remainder left after division algorithm is constant.

Let us take an example, p(x) = x^{2}+1 is a polynomial to be divided by g(x)=x.

So,( x^{2}+1)/x= x=q(x)

And r(x)=1

Clearly, the degree of remainder here is 0.

Hence, division algorithm is satisfied here.

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## 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:

**(i) **2x^{3}+x^{2}-5x+2; -1/2, 1, -2**Answer (i):**

Given, p(x) **= **2x^{3}+x^{2}-5x+2

And zeroes for p(x) are = 1/2, 1, -2

∴ p(1/2) = 2(1/2)^{3}+(1/2)^{2}-5(1/2)+2 = (1/4)+(1/4)-(5/2)+2 = 0

p(1) = 2(1)^{3}+(1)^{2}-5(1)+2 = 0

p(-2) = 2(-2)^{3}+(-2)^{2}-5(-2)+2 = 0

Hence, proved 1/2, 1, -2 are the zeroes of 2x^{3}+x^{2}-5x+2.

Now, comparing the given polynomial with general expression, we get;

∴ ax^{3}+bx^{2}+cx+d = 2x^{3}+x^{2}-5x+2

a=2, b=1, c= -5 and d = 2

As we know, if α, β, γ are the zeroes of the cubic polynomial ax^{3}+bx^{2}+cx+d , then;

α +β+γ = –b/a

αβ+βγ+γα = c/a

α βγ = – d/a.

Therefore, putting the values of zeroes of the polynomial,

α+β+γ = ½+1+(-2) = -1/2 = –b/a

αβ+βγ+γα = (1/2×1)+(1 ×-2)+(-2×1/2) = -5/2 = c/a

α β γ = ½×1×(-2) = -2/2 = -d/a

Hence, the relationship between the zeroes and the coefficients are satisfied.

**(ii)** x^{3}-4x^{2}+5x-2 ;2, 1, 1**Answer (ii):**

Given, p(x) = x^{3}-4x^{2}+5x-2

And zeroes for p(x) are 2,1,1.

∴ p(2)= 2^{3}-4(2)^{2}+5(2)-2 = 0

p(1) = 1^{3}-(4×1^{2 })+(5×1)-2 = 0

Hence proved, 2, 1, 1 are the zeroes of x^{3}-4x^{2}+5x-2

Now, comparing the given polynomial with general expression, we get;

∴ ax^{3}+bx^{2}+cx+d = x^{3}-4x^{2}+5x-2

a = 1, b = -4, c = 5 and d = -2

As we know, if α, β, γ are the zeroes of the cubic polynomial ax^{3}+bx^{2}+cx+d , then;

α + β + γ = –b/a

αβ + βγ + γα = c/a

α β γ = – d/a.

Therefore, putting the values of zeroes of the polynomial,

α +β+γ = 2+1+1 = 4 = -(-4)/1 = –b/a

αβ+βγ+γα = 2×1+1×1+1×2 = 5 = 5/1= c/a

αβγ = 2×1×1 = 2 = -(-2)/1 = -d/a

Hence, the relationship between the zeroes and the coefficients are satisfied.

**Question 2:**

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

**Answer 2:**

Let us consider the cubic polynomial is ax^{3}+bx^{2}+cx+d and the values of the zeroes of the polynomials be α, β, γ.

As per the given question,

α+β+γ = -b/a = 2/1

αβ +βγ+γα = c/a = -7/1

α βγ = -d/a = -14/1

Thus, from above three expressions we get the values of coefficient of polynomial.

a = 1, b = -2, c = -7, d = 14

Hence, the cubic polynomial is x^{3}-2x^{2}-7x+14

**Question 3:**

If the zeroes of the polynomial x^{3}-3x^{2}+x+1 are a – b, a, a + b, find a and b.

**Answer 3:**

We are given with the polynomial here,

p(x) = x^{3}-3x^{2}+x+1

And zeroes are given as a – b, a, a + b

Now, comparing the given polynomial with general expression, we get;

∴px^{3}+qx^{2}+rx+s = x^{3}-3x^{2}+x+1

p = 1, q = -3, r = 1 and s = 1

Sum of zeroes = a – b + a + a + b

-q/p = 3a

Putting the values q and p.

-(-3)/1 = 3a

a=1

Thus, the zeroes are 1-b, 1, 1+b.

Now, product of zeroes = 1(1-b)(1+b)

-s/p = 1-b^{2}

-1/1 = 1-b^{2}

b^{2} = 1+1 = 2

b = √2

Hence,1-√2, 1 ,1+√2 are the zeroes of x^{3}-3x^{2}+x+1.

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**Question 4:**

If two zeroes of the polynomial x^{4}-6x^{3}-26x^{2}+138x-35 are 2 ±√3, find other zeroes.

**Answer 4:**

Since this is a polynomial equation of degree 4, hence there will be total 4 roots.

Let f(x) = x^{4}-6x^{3}-26x^{2}+138x-35

Since 2 +√**3 **and 2-√**3 **are zeroes of given polynomial f(x).

∴ [x−(2+√3)] [x−(2-√3)] = 0

(x−2−√3)(x−2+√3) = 0

On multiplying the above equation we get,

x^{2}-4x+1, this is a factor of a given polynomial f(x).

Now, if we will divide f(x) by g(x), the quotient will also be a factor of f(x) and the remainder will be 0.

So, x^{4}-6x^{3}-26x^{2}+138x-35 = (x^{2}-4x+1)(x^{2} –2x−35)

Now, on further factorizing (x^{2}–2x−35) we get,

x^{2}–(7−5)x −35 = x^{2}– 7x+5x+35 = 0

x(x −7)+5(x−7) = 0

(x+5)(x−7) = 0

So, its zeroes are given by:

x= −5 and x = 7.

Therefore, all four zeroes of given polynomial equation are: 2+√**3** , 2-√3, −5 and 7.

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## For More Content related to Class 10 –

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