If alpha and beta are the zeros of the polynomial 4x square-16x+15, form a polynomial whose zeros are alpha by beta,beta by alpha
roots of 4x^2-16x+15=0 are a and b, (for alpha and beta)
by the properties of roots of a quadratic
a+b = -(-16/4) = 4
ab = 15/4
new roots: a/b and b/a
sum of new roots = a/b + b/a
= (a^2 + b^2)/(ab)
now a^2 + b^2 = (a+b)^2 - 2ab
= 4^2 - 2(15/4) = 17/2
so sum of new roots
= (17/2)/(15/4) = 34/15
product of new roots
= (a/b)(b/a) = 1
new equation: x^2 - 34/15 x + 1 = 0
15x^2 - 34x + 15 = 0
2nd question:
5x^2 - 4x - 8x = 0
5x^2 - 12x = 0
x(5x - 12) = 0
x = 0 or x = 12/5
I suspect a typo
Find the zeros of 5x square-4x-8x and verify the relationship between zeros and coefficient
To form a polynomial whose zeros are the reciprocals (or interchange) of the given zeros, we can use the fact that if alpha and beta are the zeros of a polynomial, then the polynomial can be factored as (x - alpha)(x - beta).
In this case, the zeros are alpha and beta, so the polynomial can be factored as:
4x^2 - 16x + 15 = (x - alpha)(x - beta)
To form a polynomial whose zeros are the reciprocals/interchange of the given zeros (beta by alpha), we need to replace alpha with beta and beta with alpha. So the polynomial becomes:
(x - beta)(x - alpha)
= (x - alpha)(x - beta)
Therefore, the polynomial with zeros given by beta by alpha is the same as the original polynomial: 4x^2 - 16x + 15.