Zeros of Polynomial Functions

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression that involves a sum of powers in one or more variables multiplied by coefficients. The general form is: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer.

Front

What are the zeros of a polynomial function?

Back

The zeros of a polynomial function are the values of \(x\) for which the function equals zero. They are also known as the roots of the polynomial.

Front

How can you find the zeros of a polynomial function?

Back

To find the zeros of a polynomial function, you can set the function equal to zero and solve for \(x\). This can be done using factoring, the quadratic formula, or numerical methods.

Front

What is the relationship between the degree of a polynomial and the number of zeros?

Back

The degree of a polynomial function indicates the maximum number of zeros it can have. For example, a polynomial of degree 3 can have up to 3 zeros.

Front

What is the Factor Theorem?

Back

The Factor Theorem states that if \(f(c) = 0\) for a polynomial function \(f(x)\), then \(x - c\) is a factor of \(f(x)\).

Front

What is the Remainder Theorem?

Back

The Remainder Theorem states that when a polynomial \(f(x)\) is divided by \(x - c\), the remainder of that division is equal to \(f(c)\).

Front

How do you determine the multiplicity of a zero?

Back

The multiplicity of a zero refers to the number of times a particular zero appears as a root of the polynomial. If a polynomial can be factored as \(f(x) = (x - r)^m g(x)\), then \(r\) is a zero with multiplicity \(m\).

Front

What is a linear factor?

Back

A linear factor of a polynomial is a factor of the form \(x - r\), where \(r\) is a root of the polynomial.

Front

What is the significance of the leading coefficient in a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It affects the end behavior of the polynomial's graph.

Front

What does it mean for a polynomial to be factored completely?

Back

A polynomial is factored completely when it is expressed as a product of its linear factors and irreducible polynomials, with no further factoring possible.