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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 =