Question

if alpha and beta are the two zeroes of polynomial 25p^2-15p+2, find a quadratic polynomial whose zeroes are 1\2a alpha and 1\2b beta

Answer 1

Answer:

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Answer 2

Apply algebra. Here is the factor theorem.

• The solutions of the polynomial are α and β.
• → There is (p-α)(p-β) as a factor.

25p²-15p+2=25(p-α)(p-β)

25p²-15p+2=25{p²-(α+β)p+αβ}

p²-3/5p+2/25=p²-(α+β)p+αβ

We have found that:

• α+β=3/5
• αβ=2/25

And this is called 'the relation between roots and coefficients'.

How do we construct such a polynomial? It is simple, just like we did before, we are gonna find sum and multiplication.

• α/2 + β/2 = 3/10
• αβ/4 = 1/50

And the polynomial is now p²-3/10p+1/50.

On making all numbers to integers, we get 50p²-15p+1.

Do not try to remember when you're doing math. It's understanding, not memorizing. I hope this helped.