(i) p(x) = x2 + 2√(2x) - 6
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 = \(\frac{-coefficient\,of\,x}{coefficient\,of\,x^2}\)
Product of zeros = √2 × - 3√2 = - 6 = - 6 = \(\frac{constant\,term}{coeffixcient\,of\,x^2}\)
(ii) q(x) = √3x2 + 10x + 7√3
⇒ √3x2 + 10x + 7√3
⇒ √3x2 + 7x + 3x + 7√3
⇒ √3x (x + \(\frac{7}{\sqrt3}\)) + 3 (x + \(\frac{7}{\sqrt3}\))
⇒ (√3x + 3) (x + \(\frac{7}{\sqrt3}\))
For zeros of Q(x), Q(x) = 0
(√3x + 3) (x + \(\frac{7}{\sqrt3}\)) = 0
X = \(\frac{-3}{\sqrt3}\), \(\frac{7}{\sqrt3}\)
Therefore zeros of the polynomial are, \(\frac{-3}{\sqrt3}\), \(\frac{7}{\sqrt3}\)
Sum of zeros = \(\frac{-3}{\sqrt3}\) + \(\frac{-7}{\sqrt3}\) = \(\frac{-10}{\sqrt3}\) = \(\frac{-coefficient\,of\,x}{coefficient\,of\,x^2}\)
Product of zeros = \(\frac{-3}{\sqrt3}\) × \(\frac{-7}{\sqrt3}\) = 7 = \(\frac{constant\,term}{coeffixcient\,of\,x^2}\)
(iii) f(x) = x2 - (√3 + 1)x + √3
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 = 1 + √3 = √3 + 1 = \(\frac{-coefficient\,of\,x}{coefficient\,of\,x^2}\)
Product of zeros = 1 × √3 = √3 = \(\frac{constant\,term}{coeffixcient\,of\,x^2}\)
(iv) g(x) = a(x2 + 1) - x(a2 + 1)
g(x) = ax2 - a2x – x + a
g(x) = ax2 - (a2 + 1)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 = \(\frac{1}{a}\), a
Therefore zeros of the polynomial are \(\frac{1}{a}\) & a
Sum of zeros
\(\frac{-coefficient\,of\,x}{coefficient\,of\,x^2}\)
Product of zeros = \(\frac{1}{a}\) × a = 1 = 1 = \(\frac{constant\,term}{coeffixcient\,of\,x^2}\)