How to find sum and product of zeros of equations

In my previous post on IB Mathematics, I have discussed how to solve a quadratic polynomial using Quadratic formula. Here I will tell you about different relationships based on sum and product of quadratic polynomial, cubic polynomial and bi-quadratic polynomials.

<img src="ib mathematics.png" alt="ib mathematics">

Quadratic:

ax2 + bx + c = 0

                                         Sum of the roots = −b/a

                                          Product of the roots = c/a

If we know the sum and product of the roots/zeros of a quadratic polynomial, then we can find that polynomial using this formula

x2 − (sum of the roots)x + (product of the roots) = 0

Cubic:

Now let us look at a Cubic (one degree higher than Quadratic):

ax3 + bx2 + cx + d=0

 if α, β and γ are the zeros of this cubic polynomial then

                                                       α+β+γ=-b/a
             
                                                   αβ+βγ+γα=c/a
           
                                                           αβγ=-d/a
If we know these relationships of polynomials then we cal also calculate the polynomial using this formula:
                         
                                       x³-(α+β+γ)x²+(αβ+βγ+γα)x-(α+β+γ)=0

Bi-Quadratic:

If we are given a bi-quadratic polynomial with degree 4 like:

                                             ax 4+bx³+cx²+dx+e=0
 
                                and its roots/zeros are α, β, γ, and δ then
 
                                           α+β+γ+δ=-b/a
 
                                          αβ+αγ+αδ+βγ+βδ+γδ=c/a
 
                                          αβγ+βγδ+γδα+αβδ=-d/a
 
                                           αβγδ=e/a
using these formulas of sum and product of zeros of polynomials, we can find a lot of relationships in zeros of polynomials. Usually, we are asked to find these types of relationships in zeros.
Question: If α and β are the zeros of polynomial x²-px+q=0 then find the following relationships.
i) 1/α+1/β   ii) α²β+αβ²   iii) α²+β²    iv) α/β+β/α   v) α³+β³
Ans: To find these relationships we convert every value either in sum (α+β)  or in the product (αβ) of zeros. For this conversion, we use following Mathematical Tricks
1)Try to take common
2) Try to take L.C.M
3) Try to make a Perfect Square
4) Use algebraic identities wherever required
If we use above steps properly, in most of the cases we are able to convert everything either in sum or in product of zeros
If we compare the given equation with std. form ax²+bx+c=0 then
                                                                       a=1, b=-p and c=q
                             sum of zeros α+β=-b/a=-(-p/a)=p

                             product of zeros    αβ=c/a=q/1=q                                                     (i) 1/α+1/β=β+α/αβ              [By L.C.M]

               
                 =p/q
(ii) α²β+αβ²  = αβ(α+β)               [By common]
                      =q.p
 (iii) α²+β² = (α²+β²+2αβ)-2αβ          [add and subtract 2αβ, make it a perfect square ]
          
          =(α+β)²-2αβ
 
          =p²-2q
(iv) α/β+β/α = β²+α²/βα          [By taking L.C.M]
 
 
we have already found β²+α² that is p²-2q so
 α/β+β/α=(p²-2q)/q

(v) (α+β)³=(α+β)³-3αβ(α+β)                [Direct algebraic identity]

                =p³-3qp
You can further read about quadratics in PDF(quadratics ) given here. There are a lot of practise questions given in this PDF
<img src="demo.png" alt="demo">

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