91Ó°ÊÓ

Polynomial zeros are the values of the variable that make the polynomial equal to zero.
For example, if we have a polynomial function like \(f(x) = x^2 - 3x + 2\), we find the zeros by solving the equation \(f(x) = 0\).
In this case, the solutions are \(x = 1\) and \(x = 2\).
These values are called the 'zeros' of the polynomial function.
This means that \(f(1) = 0\) and \(f(2) = 0\).

Zeros are important because they tell us where the function crosses the x-axis in a graph.
For any polynomial, the zeros depend on the factors that make up the polynomial.
For instance, if we have the zeros at \(1\), \(2\), and \(6\), the polynomial can be expressed in factored form as \((x-1)(x-2)(x-6) = 0\).

Notice, there can be more than one polynomial with the same zeros if we allow different constants.

An algebraic form represents a polynomial in its factored or expanded state.
Knowing how to write a polynomial in different forms helps us to understand its properties better.
For example, the polynomial \(f(x) = x^3 - 9x^2 + 26x - 24\) can be written in its factored form using its zeros (roots): \((x-1)(x-2)(x-6)\).

This conversion is crucial when solving polynomial equations or sketching graphs.
When given zeros, you can always rewrite a polynomial in its factored form.
To get the expanded form back, you simply multiply the factors.
However, remember that multiplying by a constant \(k\), such as making the polynomial \(k(x-1)(x-2)(x-6)\), does not change the zeros, but expands the polynomial differently.

Constants in polynomial equations act as multipliers that can stretch or shrink the polynomial graph.
In the general polynomial form \(k(x-1)(x-2)(x-6)\), the constant \(k\) is very significant.
It alters the polynomial but does not change the zeros, which stay at \(1\), \(2\), and \(6\).

This means \(k\) can be any non-zero value and we get different polynomial functions.
For example, if \(k\) is \(2\), then the polynomial becomes \(2(x-1)(x-2)(x-6)\).
This helps us confirm that there can be many different polynomial functions sharing the same zeros, but each shaped differently.

The constant \(k\) therefore determines the amplitude but does not move the positions of the zeros in the polynomial.