1. ## polynomial

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2. ## Re: polynomial

Hints:

a) Factor Theorem

b) Factor Theorem

c) Differentiation and Factor Theorem

3. ## Re: polynomial

This is where you consider the product of roots, which is 1.

$Since \ \alpha \ is a multiple root, then it must AT LEAST have a multiplicity of 2, we're also given that \frac{1}{\alpha} \ is a root, so if \ \alpha \ \ has a multiplicity of 3, then the roots would be \alpha , \alpha , \alpha , \frac{1}{\alpha} which would imply that \ \alpha =+-1 , \ \ but we know that can't happen from part a). Therefore \alpha only has a multiplicity of 2. Therefore we know that 3 of the roots are \alpha , \alpha,\frac{1}{\alpha}. The fourth root must also be \frac{1}{\alpha} to satisfy the product of roots.$

4. ## Re: polynomial

$Now we consider the sum of roots and the sum of roots 2 at a time. \\ \\ -A = 2 \alpha + \frac{2}{\alpha} \\ \\ B = \alpha^2 + \frac{1}{\alpha^2} + 4 \\ \\ Wtih proper subsitution , you should get the given result in the question.$

5. ## Re: polynomial

If a and b are roots of the equation x^2 + 8x -5=0, find the quadratic equation whose roots are a/b and b/a

6. ## Re: polynomial

Originally Posted by 19KANguy
If a and b are roots of the equation x^2 + 8x -5=0, find the quadratic equation whose roots are a/b and b/a
Some hints.

$\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} = \frac{74}{-5}$

$\frac{a}{b} \cdot \frac{b}{a} = 1$

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