1) f(x) =x² (x -1)⁴ (x + 5)
= x. x (x -1)(x-1)(x-1)(x-1) (x+5)
The zeros are the values of x that makes the polynomials zero
From the expression above, the zeros are: 0, 0, 1, 1, 1, 1, -5
Hence, 0 = 2 mutiplicity
1 = 4 multiplicity
-5 = 1 multiplicity
Effect
The multiplicity of a root affects the shape of the graph of a polynomial. Specifically, If a root of a polynomial has odd multiplicity, the graph will cross the x-axis at the root. If a root of a polynomial has even multiplicity, the graph will touch the x-axis at the root but will not cross the x-axis
In summary
Zero Multiplicity Effect
0 2 The graph will touch the x-axis at the root but will not cross the x-axis
1 4 The graph will touch the x-axis at the root but will not cross the x-axis
-5 1 The graph will cross the x-axis at the root.