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The zeros of a polynomial are the solutions to the equation when the polynomial is set equal to zero. Essentially, these are the x-values where the graph of the polynomial will touch or cross the x-axis. In our given exercise, these zeros are the numbers 4, -3, 3, and 0. When you have a list of zeros, you're saying this is where, if you plug these numbers into the polynomial, it equals zero.

These zeros are crucial because they provide us the foundational information needed to form the polynomial function. Without them, we would not know where the graph intersects the x-axis, which is an essential characteristic of polynomial graphs. To find the polynomial itself, we use each zero to create factors, which is the next step in the process.

To turn zeros into a polynomial function, we must express each zero as a factor. A zero of a polynomial, like our 4, -3, 3, and 0, translates into a factor using the format

• (x - Zero). For a zero of 4, the factor is (x-4).
• For -3, the factor becomes (x+3).
• Similarly, a zero of 3 turns into (x-3).
• And, the zero at 0 simply remains as a factor of x (since x = 0).

These factors are the building blocks of your polynomial. They define not only where the polynomial equation equals zero but also the roots of the equation. Together, these factors can be multiplied to start forming the polynomial equation.

It's important to note that each factor corresponds directly to one of the zeros assigned, and if one misses a factor, then the polynomial won't match the given zeros.

Simplifying a polynomial involves taking a product of factors and expanding them, which enables a more standard polynomial form that’s easier to work with. In our step-by-step exercise, starting with factors like

• (x-4),
• (x+3),
• (x-3),
• and x,

they multiply together to potentially form a complex expression. When multiplying these factors, there are a number of steps involved, especially for each pair of parentheses. First, you perform multiplications like

• x (x-4)(x+3), and then,
• you multiply the result with (x-3).

This method involves applying the distributive property, ensuring each component within one polynomial gets multiplied by each component of another. Finally, combine any like terms, leaving you with a polynomial easier to understand and use.

The polynomial from the exercise simplified to [x^4 - 4x^3 - 9x^2 + 36x.] This representation gives a clear picture of its degree (the highest power of x being 4) and makes it straightforward to use in graphing or further analysis.