Question

Find thequadratic polynomial whose sum and the product of zeroes are 3/5 and -1/3 respectively?

Answer 1

EXPLANATION.

Quadratic polynomial.

Sum of the zeroes = 3/5.

Products of the zeroes = -1/3.

As we know that,

Sum of the zeroes of the quadratic polynomial.

⇒ α + β = - b/a.

⇒ α + β = 3/5. - - - - - (1).

Products of the zeroes of the quadratic polynomial.

⇒ αβ = c/a.

⇒ αβ = -1/3. - - - - - (2).

As we know that,

Formula of quadratic polynomial.

⇒ x² - (α + β)x + αβ.

Put the values in the equation, we get.

⇒ x² - (3/5)x + (-1/3) = 0.

⇒ x² - 3x/5 - 1/3 = 0.

⇒ 15x² - 9x - 5 = 0.

                                                                                                                         

MORE INFORMATION.

Conjugate roots.

(1) = If D < 0.

One roots = α + iβ.

Other roots = α - iβ.

(2) = If D > 0.

One roots = α + √β.

Other roots = α - √β.

Answer 2

$\underline{ \sf \pmb {GIVEN \: \: - }}$

$\sf: \sf \longrightarrow \rm \: Sum \: of \: the \: zeroes = \bf\dfrac{3}{5}$

$\sf: \sf \longrightarrow \rm \: Products \: of \: the \: zeroes = \bf\dfrac{ - 1}{3}$

$\underline{ \sf \pmb {WE \: KNOW \: \: - }}$

$\sf \: Formula \: of \: quadratic \: polynomial,$

$\sf: \sf \longrightarrow \underline{\red{\boxed{\rm \: x^{2} - (sum \: of \: the \: zeroes) x + \: (products \: of \: the \: zeroes)}}}$

$\sf \: Put \: the \: values \: in \: the \: equation,$

$\begin{gathered} \sf: \sf \longrightarrow \rm \: x^{2} - \bigg(\dfrac{3}{5} \bigg) x + \:\bigg(\dfrac{ - 1}{3} \bigg) = 0\\\\\end{gathered}$

$\begin{gathered}\sf: \sf \longrightarrow \rm \: x² - \dfrac{3x}{5} - \dfrac{1}{3} = 0\\\\\end{gathered}$

$\begin{gathered} \sf: \sf \longrightarrow \bf \: 15x^² - 9x - 5 = 0\end{gathered}$