What if x is real, as i know the value of x determine as root not multiplicity of polynomial? If described polynomial don't have exponent, it's mean it's doesn't have any multiplicity or 1. Right?

When \( x \) is a real number and you want to determine if it is a root of a polynomial, you are interested in whether \( x \) makes the polynomial equal to zero when substituted for \( x \). The concept of multiplicity comes into play when considering how many times \( x \) is a root of the polynomial.

1. **Multiplicity of a Root:** The multiplicity of a root refers to the number of times that root appears in the polynomial. For example, if \( x = 2 \) is a root of the polynomial \( P(x) = (x-2)^3 \), then the multiplicity of the root \( 2 \) is 3.

2. **Polynomials without Exponents:** If a polynomial is given in a form where no explicit exponents are shown, each term is typically understood to have an exponent of 1. For example, the polynomial \( P(x) = x - 2 \) implicitly has the exponent 1 on \( x \).

3. **Multiplicity in Polynomials without Explicit Exponents:** When a polynomial does not show an explicit exponent, it means the multiplicity of each root is 1 unless specified otherwise. For instance, if you have \( P(x) = x - 3 \), the root \( x = 3 \) has a multiplicity of 1.

To summarize:
- If a polynomial does not explicitly show exponents, each term's exponent is implicitly 1.
- A root has a multiplicity of 1 if it appears only once in the polynomial.

### Examples

1. \( P(x) = (x - 2)(x - 3) \)
   - The roots are \( x = 2 \) and \( x = 3 \), each with multiplicity 1.

2. \( P(x) = (x - 4)^2 \)
   - The root \( x = 4 \) has multiplicity 2 because it appears twice as a factor.

Would you like more details or have any specific questions?

### Questions for You to Ask
1. How do you determine the multiplicity of a root in a polynomial?
2. What is the relationship between the degree of a polynomial and its roots?
3. How do you factorize a polynomial to find its roots?
4. Can a polynomial have roots with different multiplicities?
5. What is the significance of the multiplicity of a root in graphing polynomials?

### Tip
When solving polynomial equations, always check the multiplicity of roots, as it influences the shape and behavior of the graph near those roots.

Explore the concept of roots and multiplicity in polynomials. Learn how to determine if a given value is a root of a polynomial and understand the significance of multiplicity in polynomial equations. Suitable for high school students and those preparing for advanced mathematics.

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