Question

can you find two numbers whose sum is (-9) and product is 1?
○ yes
○ no
this indicates that the trinomial $x^2 - 9xy + y^2$ —select— prime.

Explanation:

Step1: Define variables for two numbers

Let the two numbers be $x$ and $y$.

Step2: Set up system of equations

We know:
$$x + y = -9$$
$$xy = 1$$

Step3: Form quadratic equation

Using the sum and product, the quadratic equation is:
$$t^2 - (x+y)t + xy = 0$$
Substitute values:
$$t^2 +9t +1 = 0$$

Step4: Check discriminant

Discriminant $D = b^2-4ac$ where $a=1, b=9, c=1$:
$$D = 9^2 - 4(1)(1) = 81 - 4 = 77$$
Since $D>0$, real solutions exist.

Step5: Analyze trinomial factorability

The trinomial $x^2 -9xy + y^2$ corresponds to the quadratic above. Since real numbers exist that multiply to 1 and add to -9, the trinomial can be factored, so it is not prime.

Answer:

Yes
is not