**Symmetric Function :**

An **algebraic expression** in a and b, which remains unchanged, when a and b are interchanged is known as **symmetric function** in a and b.

For example, *α*^{2} + *β*^{2} and *α*^{3} + *β*^{3} etc. are symmetric functions.

Symmetric function is to be expressed in terms of (*α* + *β*) and *α**β*. So, this can be evaluated for a given quadratic equation.

Some useful relations involving *α* and *β*

1. *α*^{2} + *β*^{2} = (*α* + *β*)^{2} – 2*α**β*

2. (*α* – *β*)^{2} = (*α* + *β*)^{2} – 4*α**β*

3. *α*^{2} – *β*^{2} = (*α* + *β*) (*α* – *β*) = (*α* + *β*)

4. *α*^{3} + *β*^{3} = (*α* + *β*)^{3} – 3*α**β* (*α* + *β*)

5. *α*^{3} – *β*^{3} = (*α* – *β*)^{3} + 3*α**β* (*α* – *β*)

6. *α*^{4} + *β*^{4}= [(*α* + *β*)^{2} – 2*α**β*]^{2} –2(*α**β*)^{2}

7. *α*^{4} – *β*^{4} = (*α*^{2} + *β*^{2}) (*α*^{2} – *β*^{2}) then use (1) and (3)

Example: If a and b are the zeroes of the polynomial ax^{2}+ bx + c. Find the value of (i)α–β(ii)α^{2}–β^{2}.

Solution: Since a and b are the zeroes or roots of the polynomial ax^{2} + bx + c.

⇒ a + b = – ; ab =

(i) (a – b)^{2} = (a + b)^{2} – 4ab

= = –=

a – b =

(ii) a^{2} + b^{2} = a^{2} + b^{2} + 2ab – 2ab

= (a + b)^{2} – 2ab

= – 2 =

Example: If a and b are the zeroes or roots of the quadratic polynomial ax^{2}+ bx + c. Find the value of (i) a^{2}– b^{2}(ii) a^{3}+ b^{3}.

Solution: Since a and b are the zeroes of ax^{2} + bx + c

⇒ a + b = –, ab =

(i) a^{2} – b^{2} = (a + b) (a – b)

= –

= –= –

= –

(ii) a^{3} + b^{3} = (a + b) (a^{2} + b^{2} – ab)

= (a + b) [(a^{2} + b^{2} + 2ab) – 3ab]

= (a + b) [(a + b)^{2} – 3ab]

= –

=–

=

=

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