Class 10 - Chapter 2: Polynomials

Question 1

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

NOTE: For a quadratic polynomial: the standard form is = ax² + bx + c:
Sum of zeroes(α + β) = −b/a
Product of zeroes(αβ) = c/a
(Important for board exams)

(i) x² − 2x − 8

x² − 2x − 8

By doing the middle term split, we get

Put (x - 4) = 0
x - 4 = 0
x = 4

Now Put (x + 2) = 0
x + 2 = 0
x = -2

Zeroes = 4, −2

Sum of zeroes(α + β) = 4 + (−2) = 2

−b/a = −(−2)/1 = 2

Product of zeroes(αβ) = 4 × (−2) = −8

c/a = −8/1 = −8

(ii) 4s² − 4s + 1

4s² − 4s + 1 = (2s − 1)²

Zeroes = 1/2, 1/2

Sum of zeroes(α + β) = 1/2 + 1/2 = 1

−b/a = −(−4)/4 = 1

Product of zeroes(αβ) = (1/2)(1/2) = 1/4

c/a = 1/4

(iii) 6x² − 3 − 7x

Rearrange:

6x² − 7x − 3

= 6x² − 9x + 2x − 3

= 3x(2x − 3) + 1(2x − 3)

= (3x + 1)(2x − 3)

Zeroes = −1/3, 3/2

Sum of zeroes(α + β) = −1/3 + 3/2

= (−2 + 9)/6

= 7/6

−b/a = −(−7)/6 = 7/6

Product of zeroes(αβ) = (−1/3)(3/2) = −1/2

c/a = −3/6 = −1/2

(iv) 4u² + 8u

4u(u + 2) = 0

Zeroes = 0, −2

Sum of zeroes(α + β) = 0 + (−2) = −2

−b/a = −8/4 = −2

Product of zeroes(αβ) = 0 × (−2) = 0

c/a = 0/4 = 0

(v) t² − 15

t² − 15 = 0

t = ±√15

Zeroes = √15, −√15

Sum of zeroes(α + β) = √15 − √15 = 0

−b/a = 0 ✓

Product of zeroes(αβ) = (√15)(−√15)

= −15

c/a = −15

(vi) 3x² − x − 4

3x² − x − 4

= 3x² − 4x + 3x − 4

= x(3x − 4) + 1(3x − 4)

= (x + 1)(3x − 4)

Zeroes = −1, 4/3

Sum of zeroes(α + β) = −1 + 4/3

= 1/3

−b/a = −(−1)/3 = 1/3

Product of zeroes(αβ) = (−1)(4/3)

= −4/3

c/a = −4/3

Question 2

Find a quadratic polynomial whose zeroes are the given numbers.

If the zeroes of a polynomial are α and β, then Polynomial = x² − (α + β)x + αβ

(i) Zeroes = 1/4, −1

Sum(α + β) = 1/4 − 1 = −3/4

Product(αβ) = −1/4

Polynomial:

x² + (3/4)x − 1/4

Multiplying by 4:

4x² + 3x − 1

(ii) Zeroes = √2, 1/3

Sum (α + β)= √2 + 1/3

Product (αβ)= √2/3

Polynomial:

x² − (√2 + 1/3)x + √2/3

Multiplying by 3:

3x² − (3√2 + 1)x + √2

(iii) Zeroes = 0, √5

Sum(α + β) = √5

Product(αβ) = 0

Polynomial:

x² − √5x

x(x − √5)

(iv) Zeroes = 1, 1

Sum(α + β) = 2

Product(αβ) = 1

Polynomial:

x² − 2x + 1

(x − 1)²

(v) Zeroes = −1/4, 1/4

Sum (α + β)= 0

Product (αβ)= −1/16

Polynomial:

x² − 1/16

Multiplying by 16:

16x² − 1

(vi) Zeroes = 4, 1

Sum(α + β) = 5

Product(αβ) = 4

Polynomial:

x² − 5x + 4

Is chapter me se almost Sum(α + β) and Product(αβ) se hi jyada questions puche jate hain.
Number of questions bahut type ke ban sakte hain to jyada practice karo ese questions ki.