Relation between coefficients and zeros of a quadratic polynomial

In summary, the conversation discusses a method to prove that the sum of the zeros of a quadratic polynomial is equal to -b/a and that the product of the zeros is equal to c/a. The suggested method involves comparing equations and using the quadratic formula. However, it is noted that it may be easier to simply factorize the polynomial.
  • #1
physics kiddy
135
1

Homework Statement



For any quadratic polynomial ax2+bx+c having zeros β and α
Prove that β + α = -b/a and αβ = c/a.

Homework Equations


The Attempt at a Solution



I have found a method myself to prove α+ β = -b/a. However, I could not prove αβ = c/a.

It goes like this.

If α and β are the zeros of the given polynomial.

a(α)2+b(α) + c = 0 ...... (i)

Also,

a(β)2+b(β) + c = 0 ......(ii)

Comparing (i) and (ii)

a(α)2+b(α) + c = a(β)2+b(β) + c

=> aα2-aβ2 = bβ - bα
=>a(α2-β2) = -b(α-β)
=>α2-β2/α-β = -b/a
=> α+β = -b/a

Please help me prove αβ = c/a using the same method.
 
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  • #2
physics kiddy said:

Homework Statement



For any quadratic polynomial ax2+bx+c having zeros β and α
Prove that β + α = -b/a and αβ = c/a.

Homework Equations



The Attempt at a Solution



I have found a method myself to prove α+ β = -b/a. However, I could not prove αβ = c/a.

It goes like this.

If α and β are the zeros of the given polynomial.

a(α)2+b(α) + c = 0 ...... (i)

Also,

a(β)2+b(β) + c = 0 ......(ii)

Comparing (i) and (ii)

a(α)2+b(α) + c = a(β)2+b(β) + c

=> aα2-aβ2 = bβ - bα
=>a(α2-β2) = -b(α-β)
=>α2-β2/α-β = -b/a
=> α+β = -b/a

Please help me prove αβ = c/a using the same method.
Use the quadratic formula for the two zeros of the quadratic polynomial.
 
  • #3
physics kiddy said:
Please help me prove αβ = c/a using the same method.
Do you have reason to think the same method will work?

SammyS said:
Use the quadratic formula for the two zeros of the quadratic polynomial.
It would be easier to just write down the factorization of the polynomial, I think.
 
  • #4
Given: aα²+bα + c = 0 ...... (i)
it factorizes: aα(α + b/a) + c = 0

Substitute for the bold α.
 
  • #5
Thank you very very very very much. I can't explain how happy I am to get the answer. Thanks again.
 
  • #6
SammyS said:
Use the quadratic formula for the two zeros of the quadratic polynomial.

Hurkyl said:
It would be easier to just write down the factorization of the polynomial, I think.

Yes, that's correct supposing that physics kiddy knows:
If α and β are the solutions to [itex]\text{a}x^2+bx+c=0\,,[/itex] then [itex]\text{a}x^2+bx+c=\text{a}(x-\alpha)(x-\beta)\ .[/itex]​

Of course, I agree that is a very handy thing to know!
 

FAQ: Relation between coefficients and zeros of a quadratic polynomial

What is the relationship between the coefficients and zeros of a quadratic polynomial?

The coefficients of a quadratic polynomial, represented as ax^2 + bx + c, are directly related to the zeros of the polynomial. The zeros, also known as the roots, are the values of x that make the polynomial equal to zero. The coefficients a, b, and c determine the location and nature of the zeros.

How do the coefficients affect the number of zeros in a quadratic polynomial?

The number of zeros in a quadratic polynomial is determined by the value of the discriminant, b^2 - 4ac. If the discriminant is positive, the polynomial will have two distinct real zeros. If the discriminant is zero, the polynomial will have one real zero. If the discriminant is negative, the polynomial will have no real zeros.

Can the coefficients of a quadratic polynomial be used to determine the nature of its zeros?

Yes, the coefficients can be used to determine the nature of the zeros. If the coefficient a is positive, the parabola will open upwards and the zeros will be both real and have the same sign. If a is negative, the parabola will open downwards and the zeros will be both real and have opposite signs. If a is zero, the polynomial will not be quadratic and will have one real zero.

How do the coefficients affect the location of the zeros in a quadratic polynomial?

The coefficient b, also known as the linear term, affects the location of the zeros by determining the x-coordinate of the vertex of the parabola. The vertex is located at x = -b/2a. This means that as b increases, the vertex will shift to the left, and as b decreases, the vertex will shift to the right. The coefficient c affects the y-coordinate of the vertex, which also impacts the location of the zeros.

Is there a relationship between the coefficients and the sum and product of the zeros in a quadratic polynomial?

Yes, there is a relationship between the coefficients and the sum and product of the zeros. The sum of the zeros is equal to -b/a and the product of the zeros is equal to c/a. This can be seen by expanding the polynomial (ax^2 + bx +c)(x - r)(x - s) and equating it to zero, where r and s are the zeros. This relationship is known as Vieta's formulas.

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