For h(x) = (x - 5)^{2}(x + 4)^{3} determine the

Use:
Zeroes and their Multiplicities:
Each real zero is the number that would cause the entire equation to equal zero.
The multiplicity of the zero is determined by the number of times that its associated factor appears in the polynomial.
The behavior of a graph at an x- intercept can be determined by examining the multiplicity of the zero.
For zeros with an even multiplicities, the graph touch the or are tangent to the x - axis at these x - values.
For zeros with odd multiplicities, the graphs cross or intersect the x - axis at these x - values.
A turning pointis a point of the graph changes from increasing to decreasing or decreasing to increasing .
A polynomial of degree n will have at most n- 1 turning point.
a) To list each real zero and its multiplicity.
By using each real zero is the number that would cause the entire equation to equal zero.
Also the zeros of the function are the solution of the linear factors they have given.
${\displaystyle 0=(x-5)(x+4)}$
To get,
${\displaystyle x-5=0,x+4=0}$
Solving each factor:
${\displaystyle x-5=0}$
Add on both side by 5,
To get,
${\displaystyle x=4}$.
${\displaystyle x+4=0}$
Subtract on both side by 4,
To get,
${\displaystyle x=-4}$.
To get,
The real zeroes are ${\displaystyle x=5\text{and}x=-4}$.
The multiplicities are the powers.
Then,
${\displaystyle x=5\text{with multiplicity}2}$.
${\displaystyle x=-4\text{with multiplicity}3}$.
b) To determine whether the graph crosses or touches the x-axis at each x-intercept.
The graph of the polynomial function will touch the x- axis at zeros with even multiplicities.
To get,
At ${\displaystyle x=5\text{with multiplicity}2}$
The graph of the polynomial touches the x- axis at ${\displaystyle x=5}$.
The graph will cross the x- axis at zero with odd muttiplicities.
To get,
At ${\displaystyle x=-4}$ with multiplicity 3
The graph will cross the x- axis at the ${\displaystyle x=-4}$
To get,
At ${\displaystyle x=-4}$ the graph crosses the x - axis and at ${\displaystyle x=5}$ the graph touches the x - axis at each x - intercept.
c) Maximum number of turning points on the graph:
A polynomial of degree n will have at most ${\displaystyle n-1}$ turning point.
Let ${\displaystyle h\left(x\right)={(x-5)}^2{(x-4)}^3}$.
First identify the leading term of the polynomial function if the function were expanded.
Then, identify the degree of the polynomial function.
This is the polynomial of degree 5.
The maximum number of turning points is ${\displaystyle 5-1=4}$.
Therefore,
a) The real zeros and its multiplicity are ${\displaystyle x=5}$ with multiplicity 2 and ${\displaystyle x=-4}$ with multiplicity 3.
b) At ${\displaystyle x=-4}$ the graph crosses the x-axis and at ${\displaystyle x=5}$ the graph touches the x - axis at each x- intercept.
c) The maximum number of turning point on the graph are 4.