Question

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i)x²- 2x - 8 (ii) 4s²-4s + 1 (iii) 6x²-3-7x (iv) 4u² + 8u (v)t² - 15 (vi) 3x²-x-4​

Answer 1

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(i) p(x) = x² – 2x – 8

= x² – 4x + 2x – 8

= x(x - 4) + 2(x - 4)

= (x + 2)(x - 4)

Zeroes of polynomial :

x + 2 = 0

x = -2

x - 4 = 0

x = 4

Relationship between the zeroes and the coefficients :

p(x) = x² – 2x – 8

a = 1, b = -2, c = -8

Sum of zeroes = α + β = -b/a = -(-2)/1 = 2

Product of zeroes = αβ = c/a = -8/1 = -8

(ii) p(s) = 4s² – 4s + 1

= 4s² – 2s - 2s + 1

= 2s(2s - 1) - 1(2s - 1)

= (2s - 1)(2s - 1)

Zeroes of polynomial :

2s - 1 = 0

2s = 1

s = 1/2, 1/2

Relationship between the zeroes and the coefficients :

p(s) = 4s² – 4s + 1

a = 4, b = -4, c = 1

Sum of zeroes = α + β = -b/a = -(-4)/4 = 1

Product of zeroes = αβ = c/a = 1/4

(iii) p(x) = 6x² – 3 – 7x

= 6x² – 7x - 3

= 6x² – 9x + 2x – 3

= 3x(2x - 3) + 1(2x - 3)

= (2x - 3)(3x + 1)

Zeroes of polynomial :

2x - 3 = 0

2x = 3

x = 3/2

3x + 1 = 0

3x = -1

x = -1/3

Relationship between the zeroes and the coefficients :

p(x) = 6x² – 3 – 7x

a = 6, b = -7, c = -3

Sum of zeroes = α + β = -b/a = -(-7)/6 = 7/6

Product of zeroes = αβ = c/a = -3/6 = -1/2

(iv) p(u) = 4u² + 8u

= 4u(u + 2)

Zeroes of polynomial :

4u = 0

u = 0

u + 2 = 0

u = -2

Relationship between the zeroes and the coefficients :

p(u) = 4u² + 8u

a = 4, b = 8, c = 0

Sum of zeroes = α + β = -b/a = -(8)/4 = -2

Product of zeroes = αβ = c/a = 0/4 = 0

(v) p(t) = t² – 15

= t² - (√15)²

= (t + √15)(t - √15)

Zeroes of polynomial :

t + √15 = 0

t = -√15

t - √15 = 0

t = √15

Relationship between the zeroes and the coefficients :

p(t) = t² – 15

a = 1, b = 0, c = -15

Sum of zeroes = α + β = -b/a = 0/1 = 0

Product of zeroes = αβ = c/a = -15/1 = -15

(vi) p(x) = 3x² – x – 4

= 3x² – 4x + 3x – 4

= x(3x - 4) + 1(3x - 4)

= (3x - 4)(x + 1)

Zeroes of polynomial :

3x - 4 = 0

3x = 4

x = 4/3

x + 1 = 0

x = -1

Relationship between the zeroes and the coefficients :

p(x) = 3x² – x – 4

a = 3, b = -1, c = -4

Sum of zeroes = α + β = -b/a = -(-1)/3 = 1/3

Product of zeroes = αβ = c/a = -4/3