?if alpha ,beta are 0's of quadratic polynomial alpha+ beta =24 alpha_beta =8. find the quadratic polynomial having alpha&beta as its 0's &verify the relationship between 0's -coefficient of polynomial?

Question

Answer

X^2-24x+8

Answer

Alpha +beta=24
alpha.beta=8
a+b=-b/a
ab=c/a
-b/a=24/1
b=-24 a=1
c/a=8
c=8
equation =x2-24×+8.

Answer

Answer

Quadratic Polynomial with Given Zeros:
Given that alpha and beta are the zeros of the quadratic polynomial, we know that the polynomial can be expressed as:
$ p(x) = (x - \alpha)(x - \beta) $
We are also given that:
$ \alpha + \beta = 24 $ and $ \alpha \beta = 8 $
Substitute the values of alpha and beta into the polynomial equation:
$ p(x) = (x - 24)(x - 8) $
$ p(x) = x^2 - 32x + 192 $

Relationship Between Zeros and Coefficients:
The relationship between the zeros of a quadratic polynomial and its coefficients can be explained using Vieta's formulas. For a quadratic polynomial $ ax^2 + bx + c $ with zeros alpha and beta, the following relationships hold:
1. Sum of zeros: $ \alpha + \beta = -\frac{b}{a} $
2. Product of zeros: $ \alpha \beta = \frac{c}{a} $
In our case, for the polynomial $ p(x) = x^2 - 32x + 192 $:
- Sum of zeros: $ \alpha + \beta = 24 $ which is equal to $ -\frac{-32}{1} = 32 $
- Product of zeros: $ \alpha \beta = 8 $ which is equal to $ \frac{192}{1} = 192 $
Therefore, the relationship between the zeros and coefficients of the quadratic polynomial is verified through Vieta's formulas.

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