If m and n are zeroes of the polynomial x2-x-6, then find a quadratic polynomial whose zeroes are (3a+2b) and (2a+3b).
Answers
Answer:
Step-by-step explanation:
Given If a and b are zeroes of the polynomial x2-x-6, then find a quadratic polynomial whose zeroes are (3a+2b) and (2a+3b).
Since a and b are zeroes of polynomial of the equation x^2 – x - 6 we have
Sum = a + b = -b/a = - (- 1)/1 = 1
Product = ab = c/a = - 6/1 = - 6
Sum = 3 a + 2 b + 2 a + 3 b
Sum = 5(a + b) = 5
Product = (3 a + 2 b)(2 a + 3 b)
= 6 a^2 + 4 ab + 9 ab + 6 b^2
= 6 (a^2 + b^2) + 13 ab
So we have the polynomial as
6((a +b)^2 - 2 ab) + 13 ab
6(a + b)^2 - 12 ab + 13 ab
6(a +b)^2 + ab
6(1)^2 + 1(-6)
6 - 6 = 0
The equation is x^2 - 5 x = 0
Answer:
x² - 5x = 0
Step-by-step explanation:
If a and b are zeroes of the polynomial x2-x-6, then find a quadratic polynomial whose zeroes are (3a+2b) and (2a+3b).
a & b are roots of x²-x-6
a + b = -(-1)/1 = 1 (sum of roots)
ab =-6/1 = -6 ( product of roots)
3a + 2b & 2a + 3b are roots
so sum of roots = 5a + 5b = 5(a +b) = 5 (1) = 5
Product of roots = (3a + 2b)(2a + 3b)
= 6a² + 6b² + 13ab
= 6((a + b)² - 2ab)) + 13ab
= 6 (a + b)² + ab
= 6(1)² + (-6)
= 6 - 6
= 0
sum of roots = 5 & Product of roots = 0
Hence Equation x² - 5x + 0 = 0
=> x² - 5x = 0