Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)


(i)


⇒ x2 - 4x + 2x – 8


⇒ x (x - 4) + 2 (x - 4)


For zeros of f(x), f(x) = 0


(x + 2) (x - 4) = 0


x = -2, 4


Therefore zeros of the polynomial are -2 & 4


Sum of zeros = -2 + 4 = 2 = -(-2) = 2 =


Product of zeros = -2 × 4 = -8 = -8 =


(ii)


⇒ 4s2 -2s - 2s + 1


⇒ 2s (2s - 1) -1 (2s - 1)


For zeros of g(s), g(s) = 0


(2s - 1) (2s - 1) = 0


s =


Therefore zeros of the polynomial are ,


Sum of zeros = + = 1 = -(- )= 1 =


Product of zeros = × = = =


(iii)


For zeros of h(t), h(t) = 0


t2 = 15


t = ± √15


Therefore zeros of t = √15 & -√15


Sum of zeros = √15 + (- √15) = 0 = 0 =


Product of zeros = × = -15 = -15 =


(iv) f(x) = x


⇒ 6x2 - 7x -3


⇒ 6x2 - 9x + 2x - 3


⇒ 3x(2x - 3) +1(2x - 3)


⇒ (3x + 1) (2x - 3)


For zeros of f(x), f(x) = 0


⇒ (3x + 1) (2x - 3) = 0


x =


Therefore zeros of the polynomial are


Sum of zeros = = = = =


Product of zeros = × = = =


(v) P (x) = x2 + 3√2x - √2x - 6


For zeros of p(x), p(x) = 0


⇒ x (x + 3√2) -√2 (x + 3√2) = 0


⇒ (x - √2) (x + 3√2) = 0


x = √2, -3√2


Therefore zeros of the polynomial are √2 & -3√2


Sum of zeros = √2 -3√2 = -2√2 = -2√2 =


Product of zeros = √2 × -3√2 = -6 = -6 =


(vi) q (x) = √3x2 + 10x + 7√3


⇒ √3x2 + 10x + 7√3


⇒ √3x2 + 7x + 3x + 7√3


⇒ √3x (x +) + 3 (x + )


⇒ (√3x + 3) (x +)


For zeros of Q(x), Q(x) = 0


(√3x + 3) (x +) = 0


X = ,


Therefore zeros of the polynomial are ,


Sum of zeros = + =


Product of zeros = = × = 7 =


(vii) f(x) = x2 - (√3 + 1)x + √3


f(x) = x2 - √3x - x + √3


f(x) = x(x - √3) -1(x - √3)


f(x) = (x - 1) (x - √3)


For zeros of f(x), f(x) = 0


(x - 1) (x - √3) = 0


X = 1, √3


Therefore zeros of the polynomial are 1 & √3


Sum of zeros =


Product of zeros = 1 × √3 = √3 = √3 =


(viii) g(x) = a(x2 + 1) – x(a2 + 1)


g(x) = ax2 - a2x – x + a


g(x) = ax(x - a) -1(x - a)


g(x) = (ax - 1) (x - a)


For zeros of g(x), g(x) = 0


(ax - 1) (x - a) = 0


X = , a


Therefore zeros of the polynomial are & a


Sum of zeros =


Product of zeros = × a = 1 = 1 =

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