Question

Find a quadratic polynomial with the given number as sum and product of zeroes : 1. -8/3 , 4/3

Answer 1

Answer:

x² + 8x/3 + 4/3

Note:

• The general form of a quadratic polynomial is given as ; ax² + bx + c .

• Zeros of a polynomial are the possible values of unknown (variable) for which the polynomial becomes zero .

• In order to find the zeros of a polynomial, equte it to zero.

• A quadratic polynomial has atmost two zero.

• If A and B are the zeros of s quadratic polynomial ax² + bx + c , then ;

Sum of zeros , (A+B) = -b/a

Product of zeros , (A•B) = c/a

• If A and B are the zeros of any quadratic polynomial, then it is given as ;

x² - (A+B)x + A•B .

Solution:

Here,

It is given that the sum and product of zeros of required quadratic polynomial are –8/3 and 4/3 respectively.

Let A and B be the zeros of the required quadratic polynomial.

Thus,

According to the question , we have ;

A + B = –8/3

A•B = 4/3

Now,

The required quadratic polynomial will be given as ; x² - (A+B)x + A•B

ie ; x² - (-8/3)x + 4/3

ie ; x² + 8x/3 + 4/3

Hence,

The required quadratic polynomial is:

x² + 8x/3 + 4/3 .

Moreover ,

Let's find the zeros of of the required quadratic polynomial by equating it to zero.

Thus,

=> x² + 8x/3 + 4/3 = 0

=> (3x² + 8x + 4)/3 = 0

=> 3x² + 8x + 4 = 0

=> 3x² + 2x + 6x + 4 = 0

=> x(3x + 2) + 2(3x + 2) = 0

=> (3x + 2)(x + 2) = 0

=> x = -2/3 , -2

Hence,

The zeros of required quadratic polynomial are ;

x = -2/3 , -2 .

Answer 2

$\huge\mathfrak\blue{Answer:}$

Given:

The sum of the zeroes of the quadratic polynomial is -8/3 and the product is 4/3.

To Find:

We need to find the quadratic polynomial.

Solution:

Let α and β be the two zeroes of the quadratic polynomial.

α + β = -8/3

αβ = 4/3

Now, the polynomial is :

 x² - ( α + β )x + αβ

= x² - ( -8/3 )x + 4/3

= x² + 8/3x + 4/3

Hence, the required polynomial is x² + 8/3x + 4/3.