if alpha and beta are the zeroes of the polynomial f(x)=x^2-3x-2 , find a quadratic polynomial whose zeroes are 1/2(alpha)+beta and 1/2(beta)+alpha ?
Please... i have no idea !!
Thanks :) I understood (Y)
To find a quadratic polynomial whose zeroes are 1/2(alpha)+beta and 1/2(beta)+alpha, we need to use the relationship between zeroes and coefficients of a polynomial.
Given that alpha and beta are the zeroes of the polynomial f(x) = x^2 - 3x - 2, we can use the following relationships:
Sum of the zeroes = alpha + beta
Product of the zeroes = alpha * beta
From the problem, we know that the zeroes we want are 1/2(alpha) + beta and 1/2(beta) + alpha. We can use these relationships to find these values.
Sum of the new zeroes:
(1/2 * alpha + beta) + (1/2 * beta + alpha)
Simplifying, we get:
3/2 * alpha + 3/2 * beta
Product of the new zeroes:
(1/2 * alpha + beta) * (1/2 * beta + alpha)
Simplifying, we get:
1/4 * alpha * beta + 1/2 * alpha * alpha + 1/2 * beta * beta + beta * alpha
Now that we have the sum and product of the new zeroes, we can write the quadratic polynomial.
Let's call the new polynomial g(x).
Since the sum of the new zeroes is 3/2 * alpha + 3/2 * beta, we have:
Sum of the new zeroes = -(coefficient of x) / (coefficient of x^2) = 3/2 * alpha + 3/2 * beta
Therefore, the coefficient of x in g(x) is:
-(3/2 * alpha + 3/2 * beta)
Since the product of the new zeroes is 1/4 * alpha * beta + 1/2 * alpha * alpha + 1/2 * beta * beta + beta * alpha, we have:
Product of the new zeroes = constant term / (coefficient of x^2) = 1/4 * alpha * beta + 1/2 * alpha * alpha + 1/2 * beta * beta + beta * alpha
Therefore, the constant term in g(x) is:
1/4 * alpha * beta + 1/2 * alpha * alpha + 1/2 * beta * beta + beta * alpha
Now we can write the quadratic polynomial g(x) using these coefficients:
g(x) = x^2 + (-(3/2 * alpha + 3/2 * beta)) * x + (1/4 * alpha * beta + 1/2 * alpha * alpha + 1/2 * beta * beta + beta * alpha)
Simplifying, we get the quadratic polynomial with the desired zeroes.
The zeroes of the polynomial x ^ 2 - 3 x - 2 are
[ 3 - sqrt ( 17 ) ] / 2
[ 3 + sqrt ( 17 ) ] / 2
so:
alpha = [ 3 - sqrt ( 17 ) ] / 2
beta = [ 3 + sqrt ( 17 ) ] / 2
x1 = alpha / 2 + beta =
[ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) + [ 3 + sqrt ( 17 ) ] / 2 =
[ 3 - sqrt ( 17 ) ] / 4 + 2 * [ 3 + sqrt ( 17 ) ] / ( 2 * 2 ) ] =
[ 3 - sqrt ( 17 ) ] / 4 + 2 * [ 3 + sqrt ( 17 ) ] / ( 2 * 2 ) =
[ 3 - sqrt ( 17 ) ] / 4 + [ 6 + 2 sqrt ( 17 ) ] / 4 =
[ 3 - sqrt ( 17 ) + 6 + 2 sqrt ( 17 ) ] / 4 =
[ sqrt ( 17 ) + 9 ] / 4
beta / 2 + alpha = alpha / 2 + beta =
[ 3 + sqrt ( 17 ) ] / ( 2 * 2 ) + [ 3 - sqrt ( 17 ) ] / 2 =
[ 3 + sqrt ( 17 ) ] / 4 + 2 * [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) ] =
[ 3 + sqrt ( 17 ) ] / 4 + 2 * [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) =
[ 3 + sqrt ( 17 ) ] / 4 + [ 6 - 2 sqrt ( 17 ) ] / 4 =
[ 3 + sqrt ( 17 ) + 6 - 2 sqrt ( 17 ) ] / 4 =
[ - sqrt ( 17 ) + 9 ] / 4
Now you must use Lagrange resolvents:
y = a x ^ 2 + b x + c = a ( x - x1 ) ( x - x2 )
in this case a = 1 so :
y = ( x - x1 ) ( x - x2 )
y = ( 1 / 4 )[ sqrt ( 17 ) + 9 ] * ( 1 / 4 )[ - sqrt ( 17 ) + 9 ]
y = [ x ^ 2 - 18 x + 64 ] / 16
y = x ^ 2 / 16 - 9 x / 8 + 4
x2 = beta / 2 + alpha =
[ 3 + sqrt ( 17 ) ] / ( 2 * 2 ) + [ 3 - sqrt ( 17 ) ] / 2 =
[ 3 + sqrt ( 17 ) ] / 4 + 2 * [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) ] =
[ 3 + sqrt ( 17 ) ] / 4 + 2 * [ 3 - sqrt ( 17 ) ] / ( 2 * 2 ) =
[ 3 + sqrt ( 17 ) ] / 4 + [ 6 - 2 sqrt ( 17 ) ] / 4 =
[ 3 + sqrt ( 17 ) + 6 - 2 sqrt ( 17 ) ] / 4 =
[ - sqrt ( 17 ) + 9 ] / 4