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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.
In general, a polynomial function in one variable looks like this:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 ext{where } a_n, a_{n-1},...,a_1, a_0 \text{ are constants and } n \text{ is a non-negative integer.}\]
Each term in the polynomial is called a monomial, and the degree of a polynomial is given by the highest power of the variable.
For example, in the polynomial function \( f(x) = 5x^2(x^2 - 16)(x+5) \), we have several terms with different powers of \( x \).
By analyzing these terms, we can determine the overall degree of the polynomial.
Understanding polynomial functions helps us model and solve real-world problems efficiently.

Factoring polynomials involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial.
This process is useful for finding the zeros of the polynomial efficiently.
Let's take the given polynomial:
\( f(x) = 5x^2 (x^2 - 16) (x + 5) \).
The part \( x^2 - 16 \) can be further factored using the difference of squares formula.
Difference of squares formula is:
\( a^2 - b^2 = (a - b)(a + b) \)
Applying the formula, we get:
\( x^2 - 16 = (x - 4)(x + 4) \).
Now, the polynomial becomes:
\( f(x) = 5x^2 (x - 4)(x + 4)(x + 5) \).
Factoring polynomials simplifies the process to determine its zeros and makes solving algebraic equations easier.

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero.
To find the zeros of a given polynomial, set each factor of the polynomial equal to zero and solve.
For the polynomial:
\( f(x) = 5x^2 (x - 4)(x + 4)(x + 5) \)
Set each factor to zero:
\( 5x^2 = 0 \), \( x - 4 = 0 \), \( x + 4 = 0 \), and \( x + 5 = 0 \).
This gives us the zeros:
\( x = 0 \), \( x = 4 \), \( x = -4 \), and \( x = -5 \).
The zero \( x = 0 \) has a multiplicity of 2 because the factor \( x^2 \) was squared.
The other zeros have a multiplicity of 1 since they appear only once.
Understanding the zeros of a polynomial helps in graphing the polynomial and solving equations related to the polynomial effectively.