Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients:
(i) f(x) = x² − 2x − 8
(ii) g(s) = 4s² − 4s + 1
(iii) h(t) = t² − 15
Answers
SOLUTION:
(i)
Let f(x)=x² - 2x - 8
By splitting the middle term
= x² - 4x + 2x - 8
= x(x- 4) + 2(x- 4)
= (x+2)(x - 4)
On putting f(x) = 0
(x+2)(x - 4) = 0
(x+2) = 0 or (x - 4)= 0
x = -2 or x = 4
Hence, the Zeroes of the polynomials are α = -2 and β = 4.
Verification :
Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x²
α+ β = −(−2)1
-2 + 4 = −(−2)1
2 = 2
Therefore, Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x².
Product of the zeroes(αβ) = constant term /Coefficient of x²
-2 × 4 = -8/1
-8 = -8
Therefore, Product of the zeroes(αβ) = constant term /Coefficient of x²
Hence, the relationship is verified.
(ii) Let g(s) = 4s² - 4s+1
By splitting the middle term
= 4s² - 2s - 2s + 1
= 2s(2s - 1)−1 (2s - 1)
= (2s - 1)(2s - 1)
On putting g(s) = 0
(2s - 1)(2s - 1)= 0
(2s - 1) = 0
2s = 1
s = 1/2
Hence, the Zeroes of the polynomials are α = 1/2 and β = 1/2
Verification :
Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x²
1/2+1/2= −(−4)/4
2/2 = 4/4
1 = 1
Therefore , Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x²
Product of the zeroes(αβ) = constant term /Coefficient of x²
1/2×1/2 = 1/4
1/4=1/4
Therefore, Product of the zeroes(αβ) = constant term /Coefficient of x²
Hence, the relationship is verified.
(iii)
Let h(t) = t²– 15
h(t) =t² –15
=( t)²– (√15)²
= (t +√15) (t -√15)
[a² - b² = (a +b) (a-b)]
On putting h(t) = 0
(t +√15) (t -√15) = 0
(t +√15) = 0
t = -√15
(t -√15) = 0
t = √15
Hence, the Zeroes of the polynomials are α = -√15 and β =√15.
Verification :
Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x²
-√15 + √15 = 0/1
0 = 0
Therefore, Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x².
Product of zeroes (αβ)= constant term /Coefficient of x²
−√15 × √15 = −15/1
-15 = -15
Therefore, Product of zeroes (αβ)= constant term /Coefficient of x²
Hence, the relationship verified.
HOPE THIS ANSWER WILL HELP YOU….
(-2)^2-2×(-2)-8=4+4-8