Class 10 Maths Chapter 2 Question Answers - Polynomials - 1

Q1: Find the sum and product of zeroes of 3x² - 5x + 6.

Here, p (x) = 3x² - 5x + 6
Comparing it with ax² + bx + c, we have
a = 3, b = - 5, c = 6
∴ Sum of the zeroes =

and, Product of the zeroes =

Q2: Find the sum and product of the zeroes of polynomial p (x) = 2x³ - 5x_² - 14x + 8.

Comparing p (x) = 2x³ - 5x² - 14x + 8 with ax³ + bx² + cx + d, we have
a = 2, b = –5,
c = - 14 and d = 8
∴ Sum of the zeroes =
Product of zeroes

Q3: Find a Quadratic polynomial whose zeroes are .

Sum of zeroes (S)

Product of roots (P)


Since the required Quadratic polynomial
= k(x² - Sx + P) ; where k is any real number.

= k

Thus, the required polynomial is
= k (x² - 2x - 1/4)

Q4: If α and β are the zeroes of a Quadratic polynomial x² + x - 2 then find the value of .

Comparing x² + x - 2 with ax² + bx + c, we have:
a = 1, b = 1, c = - 2

Thus,

Q5: If a and b are the zeroes of x² + px + q then find the value of .

Comparing x² + px + q with ax² + bx + c
a =1, b = p and c = q
∴ Sum of zeroes, a + b = - b/a
⇒
and αβ = c/a
⇒ αβ = q/1 = q
Now,

Thus, the value of  is

Q6: Find the zeroes of the quadratic polynomial 6x² - 3 - 7x.

We have,
= 6x² - 3 - 7x = 6x² - 7x - 3
= 6x² - 9x + 2x - 3
= 3x (2x - 3) + 1 (2x - 3)
= (3x + 1) (2x - 3)
For 6x² - 3 - 7x to be equal to zero,
either (3x + 1) = 0 or (2x - 3) = 0
⇒ 3x = - 1 or 2x = 3
⇒
Thus, the zeroes of and 3/2.

Q7: Find the zeroes of 2x² - 8x + 6.

We have,
2x² - 8x + 6 = 2x² - 6x - 2x + 6
= 2x (x - 3) - 2 (x - 3)
= (2x - 2) (x - 3)
= 2 (x - 1) (x - 3)
For 2x² - 8x + 6 to be zero,
Either, x - 1 = 0 ⇒ x = 1
or x - 3 = 0 ⇒ x = 3
∴ The zeroes of 2x² - 8x + 6 are 1 and 3.

Q8: Find the zeroes of the quadratic polynomial 3x² + 5x - 2.

We have,
p (x) = 3x² + 5x - 2
= 3x² + 6x - x - 2
= 3x (x + 2) - 1 (x + 2)
= (x + 2) (3x - 1)
For p (x) = 0, we get
Either x + 2 = 0 ⇒ x = - 2
or 3x - 1 = 0 ⇒ x = 1/3
Thus, the zeroes of 3x² + 5x - 2 are - 2 and 1/3.

Q9: If the zero of a polynomial p (x) = 3x² - px + 2 and g (x) = 4x² - q x - 10 is 2, then find the value of p and q.

∵ p (x) = 3x² - px + 2
∴ p (2) = 3 (2)² - p (2) + 2 = 0
[2 is a zero of p (x)]
or 12 - 2p + 2 or 14 - 2p = 0
or p = 7
Next g (x) = 4x² - q x - 10
∴ g (2) = 4(2)² - Q (2) - 10 = 0
[2 is a zero of g (x)]
or 4 × 4 - 2q - 10 = 0
or 16 - 2q - 10 = 0
or 6 - 2q = 0
⇒ q = 6/2 ⇒ q = 3
Thus, the required values are p = 7 and q = 3.

Q10: Find the value of ‘k’ such that the quadratic polynomial 3x² + 2kx + x - k - 5 has the sum of zeroes as half of their product.

Here, p (x) = 3x² + 2kx + x - k - 5
= 3x² + (2k + 1) x - (k + 5)
Comparing p (x) with ax² + bx + c, we have:
a = 3, b = (2k + 1),
c = - (k + 5)
∴ Sum of the zeroes

Product of the zeroes

According to the condition,
Sum of zeroes = 1/2 (product of roots)

⇒ - 2 (2k + 1) = - (k + 5)
⇒ 2 (2k + 1) = k + 5
⇒ 4k + 2 = k + 5
⇒ 4k - k = 5 - 2
⇒ 3k = 3
⇒ k = 3/3 = 1

Q11: On dividing p (x) by a polynomial x - 1 - x², the Quotient and remainder were (x - 2) and 3 respectively. Find p (x).

Here,dividend = p (x)
Divisor, g (x) = (x - 1 - x²)
Quotient, q(x) = (x - 2)
Remainder, r (x) = 3
∵ Dividend = [Divisor × Quotient] +Remainder
∴ p (x)= [g (x) × q(x)] + r (x)
= [(x - 1 - x²) (x - 2)] + 3
= [x² - x - x³ - 2x + 2 + 2x²] + 3
= 3x² - 3x - x³ + 2 + 3
= - x³ + 3x² - 3x + 5

Q12: Find the zeroes of the polynomial f (x) = 2 - x².

We have f (x)= 2 - x²
= (√2 )² - x²

Q13: Find the cubic polynomial whose zeroes are 5, 3 and - 2.

∵ 5, 3 and - 2 are zeroes of p (x)
∴ (x - 5), (x - 3) and (x + 2) are the factors of p (x)
⇒ p (x) = k (x - 5) (x - 3) (x + 2)
= k (x² - 8x + 15) (x + 2)
= k (x³ - 8x² + 15x + 2x² - 16x + 30
= k (x³ + [- 8 + 2] x² + [15 - 16] x + 30)
= k (x³ - 6x² - x + 30)
Thus, the required polynomial is k (x³ - 6x² - x + 30).

Q14: If α, β and γ be the zeroes of a polynomial p (x) such that (α + β +γ) = 3, (αβ + βγ + γα) = -10 and αβγ = - 24 then find p (x).

Here, α + β + γ = 3
αb + βγ + γα = - 10
αβγ = - 24
∵ A cubic polynomial having zeroes as α,β,γ is
p (x) = x³ - (a + b + γ) x² + (αβ + βγ + γα) x - (αβγ)
∴The required cubic polynomial is
= k {x³ - (3) x² + (- 10) x - (- 24)}
= k(x³ - 3x² - 10x + 24)

Note: If α, β and γ be the zeroes of a cubic polynomial p (x) then
p (x) = x³ - [Sum of the zeroes] x² + [Product of the zeroes taken two at a time] x - [Product of zeroes]
i.e., p (x) = k {x³ - (α + β + γ) x² + [αβ + βγ + γα] x - (αβγ).

Q15: Find all the zeroes of the polynomial 4x⁴ - 20x³ + 23x² + 5x - 6 if two of its zeroes are 2 and 3.

Here, p (x) =4x⁴ - 20x³ + 23x² + 5x - 6
Since, 2 and 3 are the zeroes of p (x),
∴ (x - 2) and (x - 3) are the factors of p(x)
⇒ (x - 2) (x - 3) is a factor of p (x)
⇒ x² - 5x + 6 is a factor of p (x)
Now, using the division algorithm for x² - 5x + 6 and the given polynomial p (x), we
get:

∴ We get (x² - 5x + 6) (4x² - 1) = p (x)
⇒ (x - 3) (x - 2) [(2x)² - (1)2] = p (x)
⇒ (x - 3) (x - 2) (2x - 1) (2x + 1) = p (x)
⇒
Thus, all the zeroes of p (x) are: 

Q16: If 1 is a zero of x³ - 3x² - x + 3 then find all other zeroes.

Here,p (x) = x³ - 3x² - x + 3
∵ 1 is a zero of p (x)
∴ (x - 1) is a factor of p (x).
Now, dividing p (x) by (x - 1), we have:

∴ p (x) = (x - 1) (x² - 2x - 3)
⇒ p (x) = (x - 1) [(x² - 3x + x - 3)]
= (x - 1) [x (x - 3) + 1 (x - 3)]
= (x - 1) [(x - 3) (x + 1)]
i.e., (x - 1), (x - 3) and (x + 1) are the factors of p (x).
⇒ 1, 3, and - 1 are the zeroes of p (x).

Q17: Find all the zeroes of 2x⁴ - 3x³ - 3x² + 6x - 2, if two of its zeroes are 1 and 1/2.

Here, p(x) = 2x⁴ - 3x³ - 3x² + 6x - 2
∵ 1 and are the zeroes of p (x)
∴ (x - 1) and  are the factors of p (x)
⇒ (x - 1) (2x - 1) is a factor of p (x)
⇒ (2x² - 3x + 1) is a factor of p (x).
Now, dividing p (x) by 2x² - 3x + 1, we get

∴ p (x) = (2x² − 3x + 1) (x² − 2)

 are the zeroes of p (x).

Q18: On dividing 4x³ - 8x² + 8x + 1 by a polynomial g (x), the Quotient and remainder were (2x² - 3x + 2) and (x + 3) respectively. Find g(x).

∵ Dividend = Divisor × Quotient + Remainder
i.e., p (x) = g (x) × Q (x) + r (x)
∴ g (x) =

Thus, the required polynomial g (x) = 2x - 1.

Q19: If α and β are the zeroes of the quadratic polynomial p (x) = kx² + 4x + 4 such that α² + β² = 24, find the value of k.

Here, p (x) = kx² + 4x + 4.
Comparing it with ax² + bx + c, we have:
a = k; b = 4; c = 4
∴ Sum of the zeroes = -b/a
⇒ α + β = -4/k
and Product of the zeroes = c/a
⇒ αβ = 4/k
∵ α² + β² = 24
∴ (α + β)² - 2αβ = 24
[∵ (x + y)² = x² + y² + 2xy ⇒ (x + y)² - 2xy = x² + y²]
⇒
⇒  
⇒ 16 − 8k − 24k² =0
⇒ 24k² + 8k − 16 = 0
⇒ (3k − 2) (k + 1) = 0
⇒ 3k − 2= 0 or k + 1 = 0
⇒ k = 2/3 or k = -1

Q20: Find the zeroes of the quadratic polynomial 6x² - 3 - 7x and verify the relationship between the zeroes and the coefficients of the polynomial.

Here, p (x) = 6x² - 3 - 7x = 6x² - 7x - 3
= 6x² - 9x + 2x - 3
= 3x (2x - 3) + 1 (2x - 3)
= (2x - 3) (3x + 1)
=
∴ Zeroes of p (x) are 3/2 and  
To verify the relationship:
Sum of the zeroes =

⇒
⇒
⇒ 7/6 = 7/6
L.H.S = R.H.S ⇒ Relationship is verified.
Product of the zeroes =

⇒
⇒
i.e., L.H.S = R.H.S ⇒ Relationship is verified.

Q21: Find the zeroes of the quadratic polynomial 5x² - 4 - 8x and verify the relationship between the zeroes and the coefficients of the polynomial.

p (x) = 5x² - 4 - 8x
= 5x² - 8x - 4
= 5x² - 10x + 2x - 4
= 5x (x - 2) + 2 (x - 2)
= (x - 2) (5x + 2)

∴ zeroes of p (x) are 2 and
Relationship Verification
Sum of the zeroes
⇒
⇒
⇒ 8/5 = 8/5

i.e., L.H.S. = R.H.S. ⇒ relationship is verified.
Product of the zeroes =
⇒
⇒
i.e., L.H.S. = R.H.S.
⇒ The relationship is verified.

Q22: Find the quadratic polynomial, the sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.

The quadratic polynomial p (x) is given by
x² - (Sum of the zeroes) x + (Product of the zeroes)
∴ The required polynomial is
= x² - [8] x + [12]
= x² - 8x + 12
To find zeroes:
∵ x² - 8x + 12 = x² - 6x - 2x + 12
= x (x - 6) - 2 (x - 6)
= (x - 6) (x - 2)
∴ The zeroes of p (x) are 6 and 2.

Q23: If one zero of the polynomial (a² - 9) x² + 13x + 6a is reciprocal of the other, find the value of ‘a’.

Here, p (x) = (a² - 9) x² + 13x + 6a
Comparing it with Ax² + Bx + C, we have:
A = (a² - 9); B = 13; C = 6
Let one of the zeroes = a
∴ The other zero = 1/α
Now, Product of the zeroes

⇒
⇒ 6a = a² − 9    ⇒ a² − 6a + 9 = 0
⇒ (a − 3)² =0   ⇒ a − 3=0
⇒ a = 3
Thus, the required value of a is 3.

Q24: If the product of zeroes of the polynomial ax² - 6x - 6 is 4, find the value of ‘a’

Here, p (x) = ax² - 6x - 6
∵ Product of zeroes =
but product of zeroes is given as 4
∴  ⇒ − 6 = 4 × a
⇒  ⇒

Thus, the required value of a is -3/2.

Q25: Find all the zeroes of the polynomial x⁴ + x³ - 34x² - 4x + 120, if two of its zeroes are 2 and - 2.

Here p (x) = x⁴ + x³ - 34x² - 4x + 120
∵ The two zeroes of p (x) are 2 and - 2
∴ (x - 2) and (x + 2) are factors of p (x)
⇒ (x - 2) (x + 2) is a factor of p (x)
⇒ x² - 4 is a factor of p (x).
Now, dividing p(x) by x² - 4, we have:

∵ Remainder = 0
∴ p (x) = (x² - 4) (x² + x - 30)
i.e., x² + x - 30 is also a factor of p (x).
∵ x² + x - 30 = x² + 6x - 5x - 30 = x (x + 6) - 5 (x + 6)
= (x + 6) (x - 5) = [x - (- 6)] [x - 5]
- 6 and 5 are also zeroes of p (x).
⇒ All the zeroes of the given polynomial are : 2, - 2, 5 and - 6

Q26: Find all the zeroes of the polynomial 2x⁴ + 7x³ - 19x² - 14x + 30, if two of its zeroes are √2 and -√2.  

P(x) = 2x⁴ + 7x³ - 19x² - 14x + 30
∵ √2 and -√2 are the two zeroes of p (x).
∴ are the factors of p (x).
⇒ i.e., x² - 2 is a factor of p (x).
Now, dividing p (x) by x² - 2, we have:

∴ p (x) = (2x² + 7x - 15) (x² - 2)
[∵ Remainder = 0]
⇒ 2x² + 7x - 15 is a factor of p (x)
∵ 2x² + 7x - 15 = 2x² + 10x - 3x - 15
= 2x (x + 5) - 3 (x + 5)
= (2x - 3) (x + 5)
=
∴ 3/2 and - 5 are zeroes of p (x)
 are the zeroes of p (x).

Q27: Find the quadratic polynomial whose zeroes are 1 and - 3. Verify the relation between the coefficients and the zeroes of the polynomial.

The given zeroes are 1 and - 3.
∴ Sum of the zeroes = 1 + (- 3) = - 2
Product of the zeroes = 1 × (- 3) = - 3
A quadratic polynomial p (x) is given by
x² - (sum of the zeroes) x + (product of the zeroes)
∴ The required polynomial is
x² - (- 2) x + (- 3)
⇒ x² + 2x - 3
Verification of relationship
∵ Sum of the zeroes

∴
⇒− 2= − 2

i.e., L.H.S = R.H.S ⇒ The sum of zeroes is verified
∵ Product of the zeroes =
∴
⇒− 3= − 3

i.e., L.H.S = R.H.S ⇒ The product of zeroes is verified.

Q28: Find the zeroes of the quadratic polynomial 4x² - 4x - 3 and verify the relation between the zeroes and its coefficients.

Here, p (x) = 4x² - 4x - 3 = 4x² - 6x + 2x - 3
= 2x (2x - 3) + 1 (2x - 3)
= (2x - 3) (2x + 1)
=
∴  are zeroes of p (x).

Verification of relationship
∵ Sum of the zeroes =

∴  
⇒ 2/2 = 1 ⇒ 1= 1
⇒
2/2 = 1 ⇒ 1 = 1
i.e., L.H.S = R.H.S ⇒ Sum of zeroes is verified
Now, Product of zeroes =

⇒  
i.e., L.H.S = R.H.S ⇒ Product of zeroes is verified.

Q29: Obtain all other zeroes of the polynomial 2x³ - 4x - x² + 2, if two of its zeroes are √2 and -√2.

p (x) = 2x³ - 4x - x² + 2
∵ √2 and -√2 are the zeroes of p (x)
∴  and are the factors of p (x)
⇒  is a factor of p (x)
⇒ x² - 2 is a factor of p (x)
Now, Dividing p (x) by (x² - 2), we have:

⇒ p (x) = (x² - 2) (2x - 1)
∴ 2x - 1 is also a factor of p (x)
i.e.,is another factor of px.
⇒ 1/2 is another zero of p (x)

Q30: Find all the zeroes of x⁴ - 3x³ + 6x - 4, if two of its zeroes are √2 and - √2.

p (x) = x⁴ - 3x³ + 6x - 4
∵ √2 and (- √2) are the zeroes of p(x)
∴ x-√2 and x-(- √2) are factors of p (x)
⇒  is a factor of p (x).
⇒ x² - 2 is a factor of p (x)
On Dividing p (x) by x² - 2, we have:

Since, remainder = 0
∴ (x² - 2) (x² - 3x + 2) = p (x)
Now, x² - 3x + 2 = x² - 2x - x + 2
= x (x - 2) - 1 (x - 2) = (x - 1) (x - 2)
i.e., (x - 1) (x - 2) is a factor of p (x)
∴ 1 and 2 are zeroes of p (x).
∴ All the zeroes of p (x) are ,√2 , - √2, 1 and 2.

Q31: Find a quadratic polynomial whose zeroes are - 4 and 3 and verify the relationship between the zeroes and the coefficients.

We know that:
P (x) = x² - [Sum of the zeroes] x + [Product of the zeroes] ...(1)
∵ The given zeroes are - 4 and 3
∴ Sum of the zeroes = (- 4) + 3 = - 1
Product of the zeroes = (- 4) × 3 = - 12
From (1), we have
x² - (- 1) x + (- 12)
= x² + x - 12    ...(2)
Comparing (2) with ax² + bx + c, we have
a = 1, b = 1, c = - 12
∴ Sum of the zeroes = -b/a
⇒ (+ 3) + (- 4) = -1/1
i.e., L.H.S = R.H.S ⇒ Sum of zeroes is verified.
Product of zeroes = c/a
⇒ 3 × (- 4) = -12/1
⇒ - 12 = - 12
i.e., L.H.S = R.H.S ⇒ Product of roots is verified.

Q32: Using division algorithm, find the quotient and remainder on dividing f (x) by g (x), where f (x) = 6x³ + 13x² + x - 2 and g (x) = 2x + 1

Here, f (x) = 6x³ + 13x² + x - 2
g (x) = 2x + 1
Now, dividing f (x) by g (x), we have:

Thus, The quotient = 3x2 + 5x - 2
remainder = 0

Q33: If the polynomial 6x⁴ + 8x³ + 17x² + 21x + 7 is divided by another polynomial 3x² + 4x + 1 then the remainder comes out to be ax + b, find ‘a’ and ‘b’.

We have:

∴ Remainder = x + 2
Comparing x + 2 with ax + b, we have
a = 1 and b = 2
Thus, the required value of a = 1 and b = 2.

Q34: If the polynomial x⁴ + 2x³ + 8x² + 12x + 18 is divided by another polynomial x² + 5, the remainder comes out to be px + q. Find the values of p and q.

We have:

∴ Remainder = 2x + 3
Comparing 2x + 3 with px + q, we have
p = 2 and q = 3

Q35: Find all the zeroes of the polynomial x³ + 3x² - 2x - 6, if two of its zeroes are - √2 and √2.

p (x) = x³ + 3x² - 2x - 6
∵ Two of its zeroes are -√2 and √2
⇒  is a factor of p (x)
⇒ x² - 2 is a factor of p (x).
Now, dividing p (x) by x² - 2 we have:

∴ p (x) = (x² - 2) (x + 3)
i.e., (x + 3) is a factor of p (x),
⇒ (- 3) is a zero of p (x)
∴All the zeroes of p (x) are - √2, √2 and - 3.

Q36: Find all the zeroes of the polynomial 2x³ + x² - 6x - 3, if two of its zeroes are -√3 and √3.

p (x) = 2x³ + x² - 6x - 3
Two of its zeroes are -√3 and √3
∴  and are factors of p (x)
i.e.,  is a factor of p (x)
⇒ x² - 3 is a factor of p (x)
Now, Dividing p (x) by x² - 3, we have:

∴ p (x) = (x² - 3) (2x + 1)
⇒  is a factor of p (x)
⇒ -1/2 is a zero of p (x)
∴ All the zeroes of p (x) are -√3 , √3 and -1/2.

Q37: Find the zeroes of the polynomial and verify the relation between the coefficients and the zeroes of the above polynomial.

The given polynomial is

∴ zeroes of the given polynomial are
Now in,
co-efficient of x² = 1
co-efficient of x = 1/6
constant term = –2
∴ Sum of zeroes  

Product of zeroes

Q38: Find the quadratic polynomial, the sum and product of whose zeroes are  respectively. Also find its zeroes.  

Sum of zeroes = √2
Product of zeroes
∵ A quadratic polynomial is given by
x² – [sum of roots] x + [Product of roots]
∴ The required polynomial is
⇒
Since =

⇒ zeroes are

Q39: If the remainder on division of x³ + 2x² + kx + 3 by x - 3 is 21, then find  the quotient and the value of k. Hence, find the zeroes of the cubic polynomial x³ + 2x² + kx - 18.

Let x³ + 2x² + kx + 3 = p(x)
∵ The divisor = x – 3
∴ p(3) = 3³ + 2 × 3² + 3k + 3
21 = 27 + 18 + 3k + 3
[∵ Remainder = 21]
⇒ 21 – 18 – 3 – 27 = 3k
⇒ –27 = 3k ⇒ k = – 9
Now, the given cubic polynomial
= x³ + 2x² – 9x + 3
since, 

∴ The required quotient = x² + 5x + 6
Now, x³ + 2x² – 9x – 18 = (x – 3) (x² + 5x + 6)
= (x – 3) (x + 3) (x + 2)
⇒ The zeroes of x³ + 2x² – 9x – 18 are 3, –3 and – 2

Q40: If a and b are zeroes of the quadratic polynomial x² – 6x + a; find the value of ‘a’ if 3α + 2β = 20.

We have quadratic polynomial = x² – 6x + a ...(1)
∵ a and b are zeroes of (1)
∴

It is given that: 3α + 2β = 20      ...(2)
Now, α +β = 6    ⇒ 2 (α+ β) = 2(6)
2α + 2β = 12      ...(3)
Subtracting (3) from (2), we have

Substituting a = 8 in α + β= 6, we get
8 +β = 6 ⇒ β = –2
Since, αβ = a
8(–2) = α ⇒ α = –16