If alpha and beta are the zeros
of the polynomial ax^2 + bx + c
then evaluateA. (alpha)^2 / beta +
(beta)^2 / alpha
B. alpha^2 .beta + alpha.beta^2
C. 1/(alpha)^4 + 1/(beta)^4.
Please work the complete solution.
idk... its wrong.. or not specific.. (abv one)
We know for any quadratic polynomial f(x)=ax^2+bx+c with roots alpha(p) and beta(q)
(x-p)(x-q)= K[x^2-(p+q)x+pq]
So we express (p+q) as -b/a and pq as c/a.....
A.)(p^2/q) + (q^2/p)=?
By simply taking LCM, we can write the above statement as
(p^3+q^3)/pq
=(p+q)(p^2+q^2-pq)[identity used]
{Now what you must understand here is that we can only substitute the values of the sum and products of the roots- so our attempt now must be towards expressing this in the form of (p+q) or pq only}
=(p+q)((p+q)^2-3pq)
On reducing by substitution-
You will obtain (3abc-b^3)/a^3
To evaluate the expressions A, B, and C using the given information that alpha and beta are the zeros of the polynomial ax^2 + bx + c, we can start by determining the relationship between the zeros and the coefficients of the polynomial.
1. Relationship between the zeros and the coefficients:
The sum of the zeros of a quadratic polynomial is given by the formula:
alpha + beta = -b/a
The product of the zeros of a quadratic polynomial is given by the formula:
alpha * beta = c/a
2. Evaluating expression A: (alpha)^2 / beta + (beta)^2 / alpha
To evaluate this expression, we can substitute the sum and product of the zeros into the expression:
(alpha)^2 / beta + (beta)^2 / alpha = [(alpha)^3 + (beta)^3] / (alpha * beta)
Using the relationship between the sum and product of the zeros, we can rewrite the expression as:
[(alpha)^3 + (beta)^3] / (alpha * beta) = [(alpha + beta)(alpha^2 - alpha * beta + beta^2)] / (alpha * beta)
Substituting -b/a for alpha + beta and c/a for alpha * beta, the expression can be simplified as:
[(alpha + beta)(alpha^2 - alpha * beta + beta^2)] / (alpha * beta) = [(-b/a)((alpha)^2 - (c/a)) / (c/a)]
Further simplifying, we get:
[(-b/a)((alpha)^2 - (c/a)) / (c/a)] = (-b)((alpha)^2 - (c/a)) / c
Therefore, the value of expression A is -b)((alpha)^2 - (c/a)) / c.
3. Evaluating expression B: alpha^2 . beta + alpha.beta^2
Using the relationship between the sum and product of the zeros, we can rewrite this expression as:
alpha^2 . beta + alpha.beta^2 = alpha.beta(alpha + beta)
Substituting -b/a for alpha + beta and c/a for alpha * beta, the expression can be simplified as:
alpha.beta(alpha + beta) = (-b/a)(c/a)
Further simplifying, we get:
(-b/a)(c/a) = -bc/a^2
Therefore, the value of expression B is -bc/a^2.
4. Evaluating expression C: 1/(alpha)^4 + 1/(beta)^4
Using the relationship between the sum and product of the zeros, we know that:
(alpha + beta)^2 = (alpha)^2 + 2alpha.beta + (beta)^2
Expanding this equation, we get:
(alpha + beta)^2 = (alpha)^2 + 2alpha.beta + (beta)^2
alpha^2 + 2alpha.beta + beta^2 = (alpha)^2 + 2alpha.beta + (beta)^2
Simplifying, we have:
alpha^2 + beta^2 = 0
Therefore, (alpha)^4 + (beta)^4 = (alpha^2 + beta^2)^2 - 2alpha^2.beta^2 = -2alpha^2.beta^2.
Substituting this relationship into expression C, we get:
1/(alpha)^4 + 1/(beta)^4 = 1/(-2alpha^2.beta^2)
Therefore, the value of expression C is 1/(-2alpha^2.beta^2).
In summary:
A. The value of expression A is -b)((alpha)^2 - (c/a)) / c.
B. The value of expression B is -bc/a^2.
C. The value of expression C is 1/(-2alpha^2.beta^2).
let the two roots be m and n
then we want
m^2/n + n^2/m
= (m^3 + n^3)/(mn)
= (m+n)(m^2 -mn + n^2)/(mn) , where m^2 + n^2 = (m+n)^2 - 2mn
= (m+n)( (m+n)^2 - 3mn)/(mn)
now from a^2 + bx + c = 0
m+n = -b/a
and mn = c/a
(m+n)( (m+n)^2 - 3mn)/(mn)
= (-b/a)( (-b/a)^2 - 3(c/a) )/(c/a)
which I reduced to
-b^3/(ca^2) - 3
check my algebra and do the others the same way