Question

Construct a quadratic polynomial whose zeroes are negatives of the zeroes of the

polynomial x


x2 − x − 12.

Answer 1

Given:

The given quadratic polynomial is :

`x^2-x-12`

To find:

The quadratic polynomial whose zeroes are negatives of the zeroes of the given polynomial.

Solution:

We have,

`x^2-x-12`

Equate the polynomial with 0 to find the zeroes.

`x^2-x-12=0`

Splitting the middle term, we get

`x^2-4x+3x-12=0`

`x(x-4)+3(x-4)=0`

`(x+3)(x-4)=0`

`x=-3,4`

The zeroes of the given polynomial are -3 and 4.

The zeroes of a quadratic polynomial are negatives of the zeroes of the given polynomial. So, the zeroes of the required polynomial are 3 and -4.

A quadratic polynomial is defined as:

`x^2-(\text{Sum of zeroes})x+\text{Product of zeroes}`

`x^2-(3+(-4))x+(3)(-4)`

`x^2-(-1)x+(-12)`

`x^2+x-12`

Therefore, the required polynomial is
`x^2+x-12`.