Question

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Answer 1

Answer:

$\frac{a(a^2-4b)(a^2-b)}{b}$

Step-by-step explanation:

Give than

α and β are zeroes of polynomial x² - ax + b.

Sum of zeroes  = -(Coefficient of x/Coefficient of x²)

          α + β       = -(-a/1)

           α + β     = a

Product of zeroes = (Constant Term/Coefficient of x²)

                 αβ        = b/1

                 αβ        = b

Now,

(α - β)² = (α + β)² - 4αβ

(α - β)² = a² - 4b

α² + β² = (α + β)² - 2αβ

α² + β² = a² - 2b

${\alpha}^2(\frac{{\alpha}^2}{\beta}-\beta)+{\beta}^2(\frac{{\beta}^2}{\alpha}-\alpha)\\\;\\={\alpha}^2(\frac{{\alpha}^2-{\beta}^2}{\beta})+{\beta}^2(\frac{{\beta}^2-{\alpha}^2}{\alpha})$

$={\alpha}^2(\frac{{\alpha}^2-{\beta}^2}{\beta})-{\beta}^2(\frac{{\alpha}^2-{\beta}^2}{\alpha})\\\;\\=\frac{{\alpha}^3({\alpha}^2-{\beta}^2)-{\beta}^3({\alpha}^2-{\beta}^2)}{\alpha\beta}\\\;\\=\frac{({\alpha}^2-{\beta}^2)({\alpha}^3-{\beta}^3)}{\alpha\beta}$

$=\frac{({\alpha}^2-{\beta}^2)(\alpha^2+\beta^2+\alpha\beta)({\alpha}-{\beta})}{\alpha\beta}\\\;\\=\frac{(\alpha-\beta)({\alpha}+{\beta})(\alpha^2+\beta^2+\alpha\beta)({\alpha}-{\beta})}{\alpha\beta}\\\;\\=\frac{(\alpha-\beta)^2({\alpha}+{\beta})(\alpha^2+\beta^2+\alpha\beta)}{\alpha\beta}$

$=\frac{(a^2-4b)a(a^2-2b+b)}{b}\\\;\\=\frac{a(a^2-4b)(a^2-b)}{b}$

Thus option 3) is correct.

Note:-

(a³ - b³) = (a - b)(a² + b² + ab)

(a² - b²) = (a + b)(a - b)

(a - b)² = (a + b)² - 4ab

Answer 2

ba(a2−4b)(a2−b)

Step-by-step explanation:

Give than

α and β are zeroes of polynomial x² - ax + b.

Sum of zeroes  = -(Coefficient of x/Coefficient of x²)

          α + β       = -(-a/1)

           α + β     = a

Product of zeroes= (Constant TermCoefficient of x²)

                αβ        = b/1

               αβ        = b

Now,

(α - β)² = (α + β)² - 4αβ

(α - β)² = a² - 4b

α² + β² = (α + β)² - 2αβ

α² + β² = a² - 2b

α^2(βα^2−β)+β^2(αβ^2−α) =α^2(βα^2−β^2)+β^2(αβ^2−α^2)

=α^2(βα^2−β^2)−β^2(αα^2−β^2) =αβα^3(α^2−β^2)−β^3(α^2−β^2)

=αβ(α^2−β^2)(α^3−β^3)

=αβ(α^2−β^2)(α^2+β^2+αβ)(α−β)

=αβ(α−β)(α+β)(α^2+β^2+αβ)(α−β)

=αβ(α−β)2(α+β)(α2+β2+αβ)

=(a^2-4b)a(a^2-2b+b)

=a(a^2-4b)

=b(a^2−4^b)a(a^2−2b+b)

=ba(a^2−4^b)(a^2−b)

Thus option 3) is correct.

Note:-

(a³ - b³) = (a - b)(a² + b² + ab)

(a² - b²) = (a + b)(a - b)

(a - b)² = (a + b)² - 4ab