Let's identify the roots and their multiplicities for each polynomial function.

1. $y = (x + 1)^5(x - 1)^3$

Root/Zero:

• The root for $(x + 1)^5$ is $x = -1$.
• The root for $(x - 1)^3$ is $x = 1$.

Multiplicity:

• The root $x = -1$ has multiplicity 5 because of the exponent on $(x + 1)$.
• The root $x = 1$ has multiplicity 3 because of the exponent on $(x - 1)$.

2. $y = (x + 4)^2(x - 3)^3$

Root/Zero:

• The root for $(x + 4)^2$ is $x = -4$.
• The root for $(x - 3)^3$ is $x = 3$.

Multiplicity:

• The root $x = -4$ has multiplicity 2.
• The root $x = 3$ has multiplicity 3.

3. $y = x(x - 3)^4(x + 4)^6$

Root/Zero:

• The root for $x$ is $x = 0$.
• The root for $(x - 3)^4$ is $x = 3$.
• The root for $(x + 4)^6$ is $x = -4$.

Multiplicity:

• The root $x = 0$ has multiplicity 1 (since there is no exponent).
• The root $x = 3$ has multiplicity 4.
• The root $x = -4$ has multiplicity 6.

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Summary:

1. Root/Zero: $x = -1$, $x = 1$
   Multiplicity: 5, 3

2. Root/Zero: $x = -4$, $x = 3$
   Multiplicity: 2, 3

3. Root/Zero: $x = 0$, $x = 3$, $x = -4$
   Multiplicity: 1, 4, 6

Would you like more details on this process or have any questions?

Related Questions:

1. How do you determine if a polynomial has real or complex roots?
2. What does it mean when a root has a higher multiplicity?
3. How does the graph of a polynomial behave near roots with different multiplicities?
4. Can a polynomial have a root with multiplicity 0? Why or why not?
5. How does the degree of a polynomial relate to the number of roots (counting multiplicity)?

Tip:

Roots with odd multiplicities (e.g., 1, 3, 5) will cross the x-axis, while roots with even multiplicities (e.g., 2, 4, 6) will "bounce" off the x-axis.