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.