To solve this problem, we analyze the given polynomial function [tex]\( f(x) = (x-1)^2(x+3)^3(x+1) \)[/tex].
1. Zero at [tex]\( x = 1 \)[/tex] and its Multiplicity:
- The term [tex]\((x-1)^2\)[/tex] indicates that [tex]\( x = 1 \)[/tex] is a root of the polynomial.
- The exponent 2 tells us the multiplicity of this root.
- Thus, the zero at [tex]\( x = 1 \)[/tex] has a multiplicity of 2.
2. Zero at [tex]\( x = -3 \)[/tex] and its Multiplicity:
- The term [tex]\((x+3)^3\)[/tex] shows that [tex]\( x = -3 \)[/tex] is a root of the polynomial.
- The exponent 3 gives the multiplicity of this root.
- Therefore, the zero at [tex]\( x = -3 \)[/tex] has a multiplicity of 3.
3. Behavior of the Graph at [tex]\( x = 1 \)[/tex]:
- A root with an even multiplicity (like 2) causes the graph to touch the x-axis at that point but not cross it.
- Hence, at [tex]\( x = 1 \)[/tex], the graph will touch, but not cross, the x-axis.
Now, let's complete the statements:
- The zero located at [tex]\( x = 1 \)[/tex] has a multiplicity of 2.
- The zero located at [tex]\( x = -3 \)[/tex] has a multiplicity of 3.
- The graph of the function will touch, but not cross, the x-axis at an [tex]\( x \)[/tex]-value of 1.
