Polynomial Functions

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression involving 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 \(a_n \neq 0\) and \(n\) is a non-negative integer.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial $$y = -4x^5 + 3x^2 - 8x + 1$$, the degree is 5.

Front

How do you determine the zeros of a polynomial?

Back

The zeros of a polynomial are the values of \(x\) for which the polynomial equals zero. They can be found by factoring the polynomial or using the quadratic formula for polynomials of degree 2.

Front

What is standard form of a polynomial?

Back

The standard form of a polynomial is when the terms are arranged in descending order of their degrees. For example, $$f(x) = x^3 - 4x^2 + 2x - 8$$ is in standard form.

Front

What is factored form of a polynomial?

Back

The factored form of a polynomial expresses it as a product of its factors. For example, $$f(x) = (x-2)(x+3)$$ is a factored form.

Front

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

Back

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

Front

What is a real zero?

Back

A real zero of a polynomial is a value of \(x\) that makes the polynomial equal to zero. For example, if \(f(x) = x^2 - 4\), then the real zeros are \(x = 2\) and \(x = -2\).

Front

How can you find the degree of a polynomial from its graph?

Back

The degree of a polynomial can often be inferred from the number of turning points in its graph. A polynomial of degree \(n\) can have at most \(n-1\) turning points.

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. It affects the end behavior of the polynomial's graph.

Front

What does it mean for a polynomial to have complex zeros?

Back

Complex zeros are solutions to the polynomial equation that are not real numbers. They occur in conjugate pairs when the polynomial has real coefficients.