Question

consider the polynomial (x-a)(x-b)and the real number line

Identify the points on the line where the polynomial is zero.

In each of the three subintervals of the line, write the sign of each factor and the sign of the product.For which
values does the polynomial possibly change signs?

Answer 1

clearly the polynomial is zero if any of its factor is zero.

So, one value of x is where x-a=0, os x=a

I expect you can find the other.

Answer 2

I understand that part, but what about the subintervals

Answer 3

Geez. We can, without loss of generality assume that a < b, so since the parabola opens upward, it dips below the x-axis between the roots.

Naturally, it changes signs at a and b.

UNLESS a=b

In that case, there is a single root, and the polynomial does not change sign, as it just touches the x-axis at x=a