Question

sum and product of zeros of a quadratic polynomial are 3 and 6 respectively then find that polynomial ​

Answer 1

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▪Given :-

For a Quadratic Polynomial

   

Sum of Zeros = 3

Product of Zeros = 6

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▪To Find :-

The Quadratic Polynomial.

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▪Key Point :-

If sum and product of zeros of any quadratic polynomial are s and p respectively,

Then,

The quadratic polynomial is given by :-

$\bf  {x}^{2}  - s \: x + p$

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▪Solution :-

Here,

Sum = s = 3

and

Product = p = 6

So,

Required Polynomial should be

$\bf{x}^{2}  - 3x + 6$

$\Large \red{\mathfrak{  \text{W}hich \:   \: is  \:  \: the  \:  \: required} }\\ \huge \red{\mathfrak{ \text{ A}nswer.}}$

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Answer 2

Answer:

Given :-

• The sum and product of zeros of a quadratic polynomial are 3 and 6 respectively.

To Find :-

• What is the polynomial.

Formula Used :-

$\clubsuit$ Quadratic Equation Formula :

$\footnotesize\mapsto \sf\boxed{\bold{\pink{x^2 - (Sum\: of\: roots)x + (Product\: of\: roots) =\: 0}}}$

Solution :-

◘ Given :

$\bigstar\: \: \: \sf\bold{\purple{Sum\: of\: roots\: (\alpha + \beta) =\: 3}}$

$\bigstar\: \: \: \sf\bold{\purple{Product\: of\: roots\: (\alpha\beta) =\: 6}}$

Hence, the required polynomial equation are :

$\small\leadsto \sf\bold{\green{x^2 - (Sum\: of\: roots)x + (Product\: of\: roots) =\: 0}}$

$\small\longrightarrow \bf{x^2 - (\alpha + \beta) + (\alpha\beta) =\: 0}$

$\small\longrightarrow \sf x^2 - (3)x + (6) =\: 0$

$\small\longrightarrow \sf x^2 - 3x + 6 =\: 0$

$\small\longrightarrow \sf\bold{\red{x^2 - 3x + 6 =\: 0}}$

${\small{\bold{\underline{\therefore\: The\: polynomial\: equation\: is\: x^2 - 3x + 6 =\: 0\: .}}}}$

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EXTRA INFORMATION :-

❒ Quadratic Equation with one variable :

✪ The general form of the equation is ax² + bx + c.

[ Note :- ◆ If a = 0, then the equation becomes to a linear equation. ]

◆ If b = 0, then the roots of the equation becomes equal but opposite in sign. ]

◆ If c = 0, then one of the roots is zero. ]

❒ Nature of the roots :

✪ Discriminant (D) = b² - 4ac. Then,

● If b² - 4ac = 0, then the roots are real & equal.

● If b² - 4ac > 0, then the roots are real & unequal.

● If b² - 4ac < 0, then the roots are imaginary & no real roots.