To write an equation of a polynomial given its zeros, you follow a structured process that uses the fact that if $x = r$ is a zero of the polynomial, then $(x - r)$ is a factor of the polynomial.

Steps to Write a Polynomial from its Zeros:

1. Identify the zeros: Suppose you are given zeros $r_1, r_2, \dots, r_n$.
2. Write factors: For each zero $r_i$, write a corresponding factor $(x - r_i)$.
3. Multiply the factors: The polynomial is the product of all these factors.
4. Expand (optional): If required, expand the product to write the polynomial in standard form.

Example:

Given Zeros:

Suppose the given zeros are $2$, $-3$, and $5$.

Step 1: Write the factors

Each zero corresponds to a factor:

• For $x = 2$, the factor is $(x - 2)$.
• For $x = -3$, the factor is $(x + 3)$.
• For $x = 5$, the factor is $(x - 5)$.

Step 2: Write the polynomial

The polynomial is the product of these factors: $P(x) = (x - 2)(x + 3)(x - 5)$

Step 3: Expand the polynomial (optional)

Now, expand the product: $P(x) = (x - 2)(x + 3)(x - 5)$ First, expand the first two factors: $(x - 2)(x + 3) = x^2 + x - 6$ Now, multiply by the third factor: $P(x) = (x^2 + x - 6)(x - 5)$ Distribute to get: $P(x) = x^3 - 5x^2 + x^2 - 5x - 6x + 30$ Simplify: $P(x) = x^3 - 4x^2 - 11x + 30$

Thus, the polynomial is: $P(x) = x^3 - 4x^2 - 11x + 30$

Key Points:

• A polynomial with degree $n$ will have $n$ zeros (some may repeat if they have multiplicity).
• The polynomial is typically written as a product of factors in the form $(x - r_i)$, where $r_i$ are the zeros.

Would you like a more complex example or further clarification on any step?

Related Questions:

1. How do you handle repeated zeros (multiplicity) when writing the polynomial?
2. How do complex numbers affect the form of the polynomial?
3. What if one of the zeros is irrational (e.g., $\sqrt{2}$)?
4. How can you use synthetic division to verify the zeros of a polynomial?
5. How does the Fundamental Theorem of Algebra relate to the number of zeros?

Tip:

Always double-check by substituting the zeros back into the polynomial to ensure they satisfy $P(x) = 0$.