How to find polynomial functions with given zeros?

Understand the Problem

The question is asking for a method to identify polynomial functions based on specific zeros provided. This generally involves using the given zeros to construct a polynomial equation by applying the fact that if 'r' is a zero, then (x - r) is a factor of the polynomial. We would multiply the factors corresponding to the zeros to form the complete polynomial.

Answer

$P(x) = (x - r_1)(x - r_2)(x - r_3)$

Steps to Solve

Identify the given zeros List the zeros provided in the problem. For example, let’s say the zeros are $r_1$, $r_2$, and $r_3$.
Construct factors from the zeros For each zero, create a factor of the polynomial. If $r_1$, $r_2$, and $r_3$ are the zeros, the corresponding factors will be: $$(x - r_1), (x - r_2), (x - r_3)$$
Multiply the factors to create the polynomial Multiply the factors together to form the polynomial. If we have three zeros, the polynomial can be expressed as: $$P(x) = (x - r_1)(x - r_2)(x - r_3)$$
Expand the polynomial (optional) If required, you can expand the polynomial to get it in standard form. This would involve using the distributive property to simplify the product.
Write the final polynomial Present the final polynomial in standard form after expansion.

The final polynomial $P(x)$ is given by: $$P(x) = (x - r_1)(x - r_2)(x - r_3)$$ After expansion, it can be expressed in standard form.

More Information

This method of constructing polynomials from their zeros is useful in various areas of algebra, particularly in polynomial factorization and solving polynomial equations.

Tips

Forgetting to include all zeros: Make sure to include all provided zeros in the construction of factors.
Incorrectly multiplying factors: Pay close attention to the multiplication process, as errors can occur when expanding the polynomial.

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