To find a quadratic polynomial whose zeroes are -4 and -5, we can use the relationship between the roots of a polynomial and its coefficients.
Let's denote the zeroes of the quadratic polynomial as [tex]\(\alpha = -4\)[/tex] and [tex]\(\beta = -5\)[/tex].
For a quadratic polynomial of the form [tex]\(ax^2 + bx + c\)[/tex]:
1. Sum of the roots ([tex]\(\alpha + \beta\)[/tex]):
[tex]\[
\alpha + \beta = -4 + (-5) = -9
\][/tex]
2. Product of the roots ([tex]\(\alpha \times \beta\)[/tex]):
[tex]\[
\alpha \beta = -4 \times -5 = 20
\][/tex]
Using these results, we can construct the quadratic polynomial in the form:
[tex]\[
ax^2 + bx + c
\][/tex]
Substituting [tex]\(a = 1\)[/tex] (since we often use the simplest form where the coefficient of [tex]\(x^2\)[/tex] is 1), we get:
[tex]\[
x^2 - (sum \ of \ roots)x + (product \ of \ roots)
\][/tex]
Thus, plugging in the sum and product of the roots:
[tex]\[
x^2 - (-9)x + 20
\][/tex]
Simplifying this, we obtain:
[tex]\[
x^2 + 9x + 20
\][/tex]
Therefore, the quadratic polynomial whose zeroes are -4 and -5 is:
[tex]\[
x^2 + 9x + 20
\][/tex]
