Explanation:
since the functional expression is already fully factorized, this is easy :
a root (or zero) of a function is a value of x that makes the function deliver 0 as result (in other words, where it intersects the x-axis).
what causes an expression like
a×b×c×d× ...
to be 0 ?
only if and when at least one of the factors (a, b, c, d, ...) is 0.
and the same applies to the products of more complex factors.
h(x) = (x-1)²(x+3)(x-4)³
let's write this by showing all factors explicitly :
h(x) = (x-1)(x-1)(x+3)(x-4)(x-4)(x-4)
yes, this is what the exponent really tells us : how often to multiply its base by itself.
again, the roots (or zeroes) of the function are all values of x that deliver h(x) = 0.
we see 6 solutions, as we have 6 factors :
x - 1 = 0, so, x = 1
x - 1 = 0, so, x = 1
x + 3 = 0, so, x = -3
x - 4 = 0, so, x = 4
x - 4 = 0, so, x = 4
x - 4 = 0, so, x = 4
while several of these roots have the same value, they are each counting.
that is why the theorem tells us that a function with the highest exponent of the variable (typically x) being n, the function has exactly n roots.
when we do the multiplications of all the factors in the given h(x), we would get for the highest exponent of x :
x²×x×x³ = x⁶.
so, the number of roots for h(x) here is 6.
what does the fact of multiple roots with the same value tell us ?
that the interception of the x-axis by the function does not do a full penetration of the x-axis but usually only touches it there.