(x-a)(x-b)
a. identify the points on the line at which the polynomial is zero
(x-a)(x-b)=0
x=a ->(a,0)
x=b->(b,0)
b. for each of three subintervals of the real number line, write the sign of each factor and the sign of the product
Since an interval was not specified for us to consider, we consider the entire domain of f which is (−∞,a) , (a,b) and (b,∞).
The split points are x=a an x=b
If both factors have the same sign (both positive, or both negative), the result is positive. Otherwise (one factor is positive, one negative), the result is negative.
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
c. at what x-value does the polynomial change signs
if x-value <0 the polynomial change signs
(-x-a)(-x-b)= x^2+ (a + b) x+ ab compare to (x-a)(x-b)=x^2 - (a + b) x + ab
more about Descartes Rule of Signs
