Question

Find the quadratic polynomial sum and product of whose zeros are -1 and-20 respectively also find the zeros of the polynomial so abstained

Answer 1

Question:

Find the quadratic polynomial sum and product of whose zeros are -1 and-20 respectively . Also find the zeros of the polynomial so abstained.

Answer:

Required quadratic polynomial : x² + x - 20

Zeros : x = -5 , 4

Note:

• A polynomial of degree 2 is called quadratic polynomial.

• A quadratic polynomial has at most two zeros.

• The possible values of variables for which the polynomial becomes zero are called its zeros.

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

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

• 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 given zeros of any quadratic polynomial then that quadratic polynomial is given by : x² - (A+B)x + A•B .

Solution:

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

Now,

According to the question,

Sum of the zeros = -1

ie ; A + B = -1

Also,

Product of the zeros = -20

ie; A•B = -20

Now,

The quadratic polynomial sum and product of whose zeros are -1 and -20 respectively will be ;

x² - (A+B)x + A•B

ie; x² - (-1)x + (-20)

ie; x² + x - 20.

Hence,

The required quadratic polynomial is :

x² + x - 20 .

Now,

In order to find the zeros of the obtained quadratic polynomial , let's equate it to zero.

Thus,

=> x² + x - 20

=> x² + 5x - 4x - 20

=> x(x+5) - 4(x+5) = 0

=> (x + 5)(x - 4) = 0

=> x = -5 , 4

Hence,

The zeros of the obtained quadratic polynomial are : x = - 5 , 4 .

Answer 2

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

Given:

The sum of the zeroes of the quadratic polynomial is -1 and their product is -20.

To Find:

We need to find the quadratic polynomial.

Solution:

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

Now, it is given that the sum (α + β) is -1

and product (αβ) is -20.

So, the quadratic polynomial will be:

x^2 - (α + β)x + (αβ)

= x^2 - (-1)x + (-20)

= x^2 + 1x -20

Hence, the required polynomial is x^2 + 1x -20.

Now, we need to find the zeroes of this polynomial.

x^2 + x -20

By splitting the middle term we get,

x^2 + 5x - 4x -20

= x(x + 5) -4(x + 5)

= (x - 4) (x + 5)

Hence, the zeroes of the polynomial are 4 and -5.