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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) x²–2x –8
Answer (i):
x²– 4x+2x–8 = x(x–4)+2(x–4) = (x-4)(x+2)

Therefore, zeroes of polynomial equation x²–2x–8 are (4, -2)

Sum of zeroes = 4–2 = 2 = -(-2)/1 = -(Coefficient of x)/(Coefficient of x²)
Product of zeroes = 4×(-2) = -8 =-(8)/1 = (Constant term)/(Coefficient of x²)

(ii) 4s²–4s+1
Answer (ii):
4s²–2s–2s+1 = 2s(2s–1)–1(2s-1) = (2s–1)(2s–1)

Therefore, zeroes of polynomial equation 4s²–4s+1 are (1/2, 1/2)
Sum of zeroes = (½)+(1/2) = 1 = -4/4 = -(Coefficient of s)/(Coefficient of s²)
Product of zeros = (1/2)×(1/2) = 1/4 = (Constant term)/(Coefficient of s² )

(iii) 6x²–3–7x
Answer (iii):
6x²–7x–3 = 6x² – 9x + 2x – 3 = 3x(2x – 3) +1(2x – 3) = (3x+1)(2x-3)

Therefore, zeroes of polynomial equation 6x²–3–7x are (-1/3, 3/2)

Sum of zeroes = -(1/3)+(3/2) = (7/6) = -(Coefficient of x)/(Coefficient of x²)
Product of zeroes = -(1/3)×(3/2) = -(3/6) = (Constant term) /(Coefficient of x² )

(iv)  4u²+8u
Answer (iv):
4u(u+2)

Therefore, zeroes of polynomial equation 4u² + 8u are (0, -2).

Sum of zeroes = 0+(-2) = -2 = -(8/4) = = -(Coefficient of u)/(Coefficient of u²)
Product of zeroes = 0×-2 

(v)  t²–15
Answer (v):
t² = 15 or t = ±√15

Therefore, zeroes of polynomial equation t² –15 are (√15, -√15)

Sum of zeroes =√15+(-√15) = 0= -(0/1)= -(Coefficient of t) / (Coefficient of t²)
Product of zeroes = √15×(-√15) = -15 = -15/1 = (Constant term) / (Coefficient of t² )

(vi) 3x²–x–4
Answer (vi):
3x²–4x+3x–4 = x(3x-4)+1(3x-4) = (3x – 4)(x + 1)

Therefore, zeroes of polynomial equation3x² – x – 4 are (4/3, -1)

Sum of zeroes = (4/3)+(-1) = (1/3)= -(-1/3) = -(Coefficient of x) / (Coefficient of x²)
Product of zeroes=(4/3)×(-1) = (-4/3) = (Constant term) /(Coefficient of x² )

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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:-
x²–(α+β)x +αβ = 0
x²–(1/4)x +(-1) = 0
4x²–x-4 = 0

Thus,4x²–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:-
x²–(α+β)x +αβ = 0
x² –(√2)x + (1/3) = 0
3x²-3√2x+1 = 0

Thus, 3x²-3√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:-
x²–(α+β)x +αβ = 0
x²–(0)x +√5= 0

Thus, x²+√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:-
x²–(α+β)x +αβ = 0
x²–x+1 = 0

Thus , x²–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:-
x²–(α+β)x +αβ = 0
x²–(-1/4)x +(1/4) = 0
4x²+x+1 = 0

Thus,4x²+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:-
x²–(α+β)x+αβ = 0
x²–4x+1 = 0

Thus, x²–4x+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) = x³-3x²+5x–3 , g(x) = x²–2
Answer (i):
Given,
Dividend = p(x) = x³-3x²+5x–3
Divisor = g(x) = x²– 2

(ii) p(x) = x⁴-3x²+4x+5 , g(x) = x²+1-x
Answer (ii):
Dividend = p(x) = x⁴ – 3x² + 4x +5
Divisor = g(x) = x² +1-x

(iii) p(x) =x⁴–5x+6, g(x) = 2–x²
Anwer (iii):
Given,

Dividend = p(x) =x⁴ – 5x + 6 = x⁴ +0x²–5x+6
Divisor = g(x) = 2–x² = –x²+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) t² – 3, 2t⁴ + 3t³ – 2t²– 9t – 12
(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2
(iii) x² + 3x + 1, x⁵ – 4x³ + x² + 3x + 1

Answer 2:

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Question 3:
Obtain all other zeroes of 3x⁴+6x³-2x²-10x-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) = x²-(5/3) = 0
(3x²−5)=0, is a factor of given polynomial f(x).

Now, when we will divide f(x) by (3x²−5) the quotient obtained will also be a factor of f(x) and the remainder will be 0.

Therefore, 3x⁴ +6x³ −2x² −10x–5 = (3x² –5)(x²+2x+1)
Now, on further factorizing (x²+2x+1) we get,

x²+2x+1 = x²+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 x³-3x²+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) = x³-3x²+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

∴ x³-3x²+x+2 = g(x)×(x-2) + (-2x+4)
x³-3x²+x+2-(-2x+4) = g(x)×(x-2)

Therefore, g(x) × (x-2) = x³-3x²+x+2
Now, for finding g(x) we will divide x³-3x²+x+2 with (x-2)

Therefore, g(x) = (x²–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, 3x²+3x+3 is a polynomial to be divided by 3.
So, (3x²+3x+3)/3 = x²+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)=x²+x is a polynomial to be divided by g(x)=x.

So, (x²+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) = x²+1 is a polynomial to be divided by g(x)=x.

So,( x²+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) 2x³+x²-5x+2; -1/2, 1, -2
Answer (i):
Given, p(x) = 2x³+x²-5x+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 2x³+x²-5x+2.
Now, comparing the given polynomial with general expression, we get;

∴ ax³+bx²+cx+d = 2x³+x²-5x+2
a=2, b=1, c= -5 and d = 2

As we know, if α, β, γ are the zeroes of the cubic polynomial ax³+bx²+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) x³-4x²+5x-2 ;2, 1, 1
Answer (ii):
Given, p(x) = x³-4x²+5x-2
And zeroes for p(x) are 2,1,1.

∴ p(2)= 2³-4(2)²+5(2)-2 = 0
p(1) = 1³-(4×1² )+(5×1)-2 = 0

Hence proved, 2, 1, 1 are the zeroes of x³-4x²+5x-2
Now, comparing the given polynomial with general expression, we get;

∴ ax³+bx²+cx+d = x³-4x²+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 ax³+bx²+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 ax³+bx²+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 x³-2x²-7x+14

Question 3:
If the zeroes of the polynomial x³-3x²+x+1 are a – b, a, a + b, find a and b.

Answer 3:
We are given with the polynomial here,
p(x) = x³-3x²+x+1

And zeroes are given as a – b, a, a + b
Now, comparing the given polynomial with general expression, we get;

∴px³+qx²+rx+s = x³-3x²+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-b²
-1/1 = 1-b²
b² = 1+1 = 2
b = √2

Hence,1-√2, 1 ,1+√2 are the zeroes of x³-3x²+x+1.

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Question 4:
If two zeroes of the polynomial x⁴-6x³-26x²+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) = x⁴-6x³-26x²+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,
x²-4x+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, x⁴-6x³-26x²+138x-35 = (x²-4x+1)(x² –2x−35)
Now, on further factorizing (x²–2x−35) we get,

x²–(7−5)x −35 = x²– 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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