Question

Find a quadratic polynomial whose sum and product of zeroes are 5 and 2 respectively.​

Answer 1

Given,

Sum of  zeroes of the polynomial = 5

Product of the zeroes of the polynomial = 2

To find,

Finding the quadratic equation with the given sum and product of zeroes.

Solution,

We can simply solve this problem by using the properties of zeroes of a quadratic equation.

let,

A quadratic polynomial is of the form p(x): ax²+bx+c, where a≠0.

and,

The quadratic equation having roots α, β, is x² - (α + β)x + αβ = 0

where a and c are the coefficients of x², x. c is the constant term.

(a is the coefficient of x² and is also called the quadratic coefficient

b is the coefficient of x and is also called the linear coefficient and

c is the constant term)

We know,

• The sum of the zeroes of a quadratic equation is α + β = -b/a = - Coefficient of x/ Coefficient of x2.
• The product of the zeroes of the quadratic equation is αβ = c/a = Constant term/ Coefficient of x2

Thus,

α + β = -b/a = 5

αβ = c/a = 2

now,   x² - (α + β)x + αβ = 0 and now puting the values of (α + β) and αβ.

x² - 5x + 2= 0

So, the required quadratic polynomial with sum and product of zeroes are 5 and 2 respectively is x² - 5x + 2= 0

Hence, the quadratic polynomial is x² - 5x + 2= 0.

Answer 2

Answer:

The quadratic polynomial is $x^{2} -5x + 2$.

Explanation:

What is a quadratic Polynomial ?

A polynomial whose highest degree monomial is of the second degree is said to be quadratic. A second-order polynomial is another name for a quadratic polynomial. Accordingly, at least one of the variables must be raised to the power of 2, and the powers of the remaining variables must be more than or equal to two but less than -1.

For a quadratic Polynomial $ax^{2} +bx + c$  the roots lets say $\alpha \ and\ \beta$, then

$\alpha +\beta = \frac{-b}{a}$ and

$\alpha \beta = \frac{c}{a}$ .

The sum of roots is 5

The product of roots is 2

The roots be $\alpha \ and\ \beta$ ,

$\alpha +\beta = 5$

⇒ $\frac{-b}{a} = 5.$

⇒ b = -5a

again,

⇒ $\alpha \beta = 2$

⇒ $\frac{c}{a} = 2$

⇒ c = 2a

Therefore the quadratic polynomial is

$ax^{2} -5ax + 2a = 0$

⇒ $x^{2} -5x + 2 = 0$

To learn more about Quadratic Polynomials refer to :

https://brainly.in/question/9688277?referrer=searchResults

https://brainly.in/question/23382810?referrer=searchResults

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