Explain the connection between factors of a polynomial, zeros of a polynomial function, and solutions of a polynomial equation.
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any polynomial which divides evenly into p(x)
Just like factors of an integer, right?
What is the definition of a "factor?"
a number or quantity that when multiplied with another produces a given number or expression.
so in the product ab a and b are called what?
yes, variables represent numbers, right?
In this^ frame work we may replace a, b, and c with polynomials because a polynomial is simply a sum of monomials.
Let's say our given polynomial can be written as a product. Then each of the factors can be called a number or quantity that when multiplied with another produces a given number or expression.
Now the "zeros of a polynomial function" is when we replace c in ab =c with 0.
We want ab = 0 to be a true statement
This can only be true if and only if a = 0 or b = 0.
This^ is called the "zero-product property"
Now you know why I made such a big deal out of the definition of a "factor" :)
And, finally the solutions of a polynomial equation are those values of the variable that make the above equation a true statement.