Certain relations involving α and β can also be determined from α+β andαβ even when we do not know α and β distinctly. Such relations are usually said to be symmetric. They are symmetric in the sense that if α and α are interchanged, either the relation remains the same or is multiplied by -1

Example – If α ≠ β, determine whether or not each of the following is symmetric

(a) α+β (b) αβ (c) α^{2} + β^{2} (d) α^{2} – β^{2} (e) 3α +2β (f) α^{3} + β^{3}

Solution

- α+β = β +α , therefore α+β is symmetric
- αβ =βα, therefore αβ is symmetric
- α
^{2}+ β^{2}= β^{2}+ α^{2}, therefore α^{2}+ β^{2}is symmetric - α
^{2}– β^{2}= – (β^{2}– α^{2}), thereforeα^{2}– β^{2}is symmetric - 3α +2β ≠ 3β + 2α since α ≠ β, therefore 3α +2β is not symmetric
- α
^{3}+ β^{3}= β^{3}+ α^{3, }therefore α^{3}+ β^{3}is symmetric

**EVALUATION (USE THE BOX AT THE BOTTOM TO POST YOUR ANSWER FOR DISCUSSION AND APPRAISAL)**

- Find the quadratic equation whose roots are ½ and 5
- Find the sum & product of roots of the equation 3x
^{2}– 5x – 2 = 0

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