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?

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

To identify the points on the real number line where the polynomial (x-a)(x-b) is zero, we need to find the values of x that make the polynomial equal to zero.

First, let's set the polynomial equal to zero and solve for x:

(x - a)(x - b) = 0

Since the product of two factors is zero if and only if at least one of the factors is zero, we can set each factor equal to zero and solve for x:

x - a = 0 or x - b = 0

This gives us two possible values for x: x = a and x = b. So, the polynomial is zero at x = a and x = b.

Now let's consider the three possible subintervals of the real number line: (-∞, a), (a, b), and (b, ∞).

For each subinterval, we will analyze the signs of the factors (x - a) and (x - b) and the sign of their product.

1. Subinterval (-∞, a):
- For this subinterval, we need to determine the sign of (x - a) and (x - b).
- Since a is greater than any value in the interval (-∞, a), (x - a) is negative for all x in this subinterval.
- Similarly, since b is greater than any value in the interval (-∞, a), (x - b) is negative for all x in this subinterval.
- The product of two negative numbers is positive, so the sign of the product is positive.

2. Subinterval (a, b):
- For this subinterval, we need to determine the sign of (x - a) and (x - b).
- Since a is less than any value in the interval (a, b), (x - a) is positive for all x in this subinterval.
- Similarly, since b is greater than any value in the interval (a, b), (x - b) is negative for all x in this subinterval.
- The product of a positive and a negative number is negative, so the sign of the product is negative.

3. Subinterval (b, ∞):
- For this subinterval, we need to determine the sign of (x - a) and (x - b).
- Since a is less than any value in the interval (b, ∞), (x - a) is positive for all x in this subinterval.
- Similarly, since b is less than any value in the interval (b, ∞), (x - b) is positive for all x in this subinterval.
- The product of two positive numbers is positive, so the sign of the product is positive.

Therefore, the polynomial (x - a)(x - b) possibly changes signs in the subinterval (a, b).

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.

I understand that part, but what about the subintervals