When given x equals a zero how do you write the factored polynomial

When a polynomial has a root \( x = a \), this means that \( (x - a) \) is a factor of the polynomial.

For example, if \( x = 3 \) is a zero of a polynomial, then \( (x - 3) \) is one of the factors. Similarly, if \( x = -2 \) is a zero, then \( (x + 2) \) is a factor.

### Process of Writing the Factored Polynomial:
1. **Identify the zero**: If you know that \( x = a \) is a zero of the polynomial, then \( (x - a) \) is a factor.
2. **For multiple zeros**: If you are given multiple zeros \( x = a_1, a_2, a_3, \dots \), the corresponding factored form will be \( (x - a_1)(x - a_2)(x - a_3) \dots \).
3. **For repeated zeros**: If a zero occurs multiple times, it should be repeated in the factorization. For example, if \( x = 2 \) is a zero with multiplicity 2, it will appear as \( (x - 2)^2 \).

#### Example:
If the polynomial has zeros \( x = 2 \), \( x = -3 \), and \( x = 4 \), then the factored form of the polynomial will be:
\[
(x - 2)(x + 3)(x - 4)
\]

If you are given just a single zero \( x = 0 \), the factor is \( x \), and the factored form is \( x \times (\text{other factors}) \).

Would you like a more detailed example or help with a specific polynomial?

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Here are 5 related questions for deeper understanding:
1. How do you factor a polynomial with complex roots?
2. What happens if the given zero has a multiplicity greater than one?
3. Can you factor a cubic polynomial if one of its roots is zero?
4. How do you factor a polynomial when only some roots are known?
5. How do you derive the factored form from a given polynomial's expanded form?

**Tip**: Always check whether the degree of the polynomial matches the number of roots (including multiplicities) to ensure that all factors are accounted for.

When given x equals a zero how do you write the factored polynomial?

Learn how to write the factored form of a polynomial given its zeros. This tutorial covers key polynomial concepts, such as identifying factors based on zeros and handling multiple and repeated zeros. Suitable for students in Grades 9-12.

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