Question

if alpha and beta are the zeroes of The polynomial p(x) = 3x²+2x+1, then form a polynomial whose zeroes are 1-alpha÷1+beta, 1-beta÷1+alpha.




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Answer 1

If alpha and beta are the zeroes of the polynomial 3x2-5x+8, then what is the alpha^2+beta^2?

Sorry, I don’t like alpha and beta; I’ll use p and q instead- heck, they’re quicker to type for a start!

A: Our original quadratic expression is; 3x2−5x+83x2−5x+8

Now, if p and q are the roots, then:

3(x−p)(x−q)=03(x−p)(x−q)=0

Multiplying out the terms, we have:

B: 3x2−3(p+q)x+3pq=03x2−3(p+q)x+3pq=0

But, A and B are the same equation, so:

−3(p+q)=−5→p+q=53−3(p+q)=−5→p+q=533pq=8→pq=833pq=8→pq=83

Now, (p+q)2=p2+2pq+q2(p+q)2=p2+2pq+q2

→p2+q2=(p+q)2−2pq→p2+q2=(p+q)2−2pq

We know the values of p + q (from 1.) and pq (from 2.), thus:

p2+q2=(53)2−283p2+q2=(53)2−283

=259−163=259−163

=259−489=259−489

=−239

Answer 2

Step-by-step explanation:

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