find a quadratic polynomial the sum and product of whose zeroes are -8/3 and 4/3 respectively also find the zeroes of polynomial by factorisation
Answers
Answer:
alpha+bheta=-8/3
alpha×bheta=4/3
Step-by-step explanation:
k(x^2-(alpha+bheta)+alpha×bheta
k(x^2-(-8/3)x+4/3)
k(3x^2+8/3x+4/3)
k(3x^2+8x+4/3)
3k(3x^2+8x+4/3)
the equation is 3x^2+8x+4.
3x^2+8x+4
=3x^2+6x+2x+4
=3x(x+2) 2(x+2)
=(x+2) (3x+2)
x=-2,-2/3
3x² + 8x + 4 = 0 is the quadratic polynomial while -2 and -2/3 are the zeroes.
Step-by-step explanation:
Given: Sum of zeroes (α + β) = -8/3
Product of zeroes (αβ) = 4/3
To Find: The quadratic polynomial and its zeroes
Solution:
- Finding the quadratic polynomial
Let α and β be the zeroes of the quadratic polynomial such that
(α + β) = -8/3 and (αβ) = 4/3
therefore, the equation can be written as,
x² - (α + β)x + (αβ) = 0
⇒ x² - (-8/3)x + (4/3) = 0
⇒ x² - (-8/3)x + (4/3) = 0
⇒ 3x² + 8x + 4 = 0
This is the quadratic polynomial whose zeroes are α and β
- Finding zeroes of the polynomial
Now, we have 3x² + 8x + 4 = 0 such that,
⇒ 3x² + 6x + 2x + 4 = 0
⇒ 3x (x + 2) + 2 (x + 2) = 0
⇒ (3x + 2) (x + 2) = 0
⇒ x = -2, -2/3
Therefore, zeroes are α = -2 and β = -2/3
Hence, 3x² + 8x + 4 = 0 is the quadratic polynomial while -2 and -2/3 are the zeroes.