Question

If α and β are the zeroes of the quadratic polynomial f(x)=ax2+bx+c
then evaluate,a(α²/β+β²/α)+b(α/β+β/α)

Answer 1

Given: α and β are the zeroes of the quadratic polynomial f(x)=ax2+bx+c.

To find:  Evaluate,a(α²/β+β²/α)+b(α/β+β/α)

Solution:

• As we have given the two degree polynomial, and α and β are roots of it, so:

         α+β = -b/a and αβ = c/a

         α+β = -b/a

• Now, using these values, we can evaluate a(α²/β+β²/α)+b(α/β+β/α)

• Lets simplify the terms, we get:

         a( α³+β³)/αβ  +  b(α²+β²)/αβ

         a ( α+β)(α²+β²-αβ)/αβ    +    b(α²+β² +2αβ - 2αβ)/αβ

        {a{ ( -b/a)(b²-2ac)/a² - c/a} / c/a}+ {b(α+β)²- 2αβ/ αβ}

        a(3abc - b³/a²c) + b{b²/a² -2c/a / c/a}

        a(3abc - b³/a²c) + b{b²-2ca/ac }

        (3abc - b³/ac) + (b³-2abc/ac)

        (3abc -2abc -b³ +b³)/ac

         abc/ac

        b

Answer:

  So, the final answer of a(α²/β+β²/α)+b(α/β+β/α) is b.