Given:
The given quadratic polynomial is :
[tex]x^2-x-12[/tex]
To find:
The quadratic polynomial whose zeroes are negatives of the zeroes of the given polynomial.
Solution:
We have,
[tex]x^2-x-12[/tex]
Equate the polynomial with 0 to find the zeroes.
[tex]x^2-x-12=0[/tex]
Splitting the middle term, we get
[tex]x^2-4x+3x-12=0[/tex]
[tex]x(x-4)+3(x-4)=0[/tex]
[tex](x+3)(x-4)=0[/tex]
[tex]x=-3,4[/tex]
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:
[tex]x^2-(\text{Sum of zeroes})x+\text{Product of zeroes}[/tex]
[tex]x^2-(3+(-4))x+(3)(-4)[/tex]
[tex]x^2-(-1)x+(-12)[/tex]
[tex]x^2+x-12[/tex]
Therefore, the required polynomial is [tex]x^2+x-12[/tex].
