5.3 Polynomial Functions and End Behavior

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 P(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 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. It determines the polynomial's end behavior and the number of roots.

Front

How do you determine the leading term of a polynomial?

Back

The leading term of a polynomial is the term with the highest degree. It is found by identifying the term with the largest exponent in the polynomial.

Front

What is the end behavior of a polynomial function?

Back

The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term.

Front

What does it mean if a polynomial has an even degree?

Back

If a polynomial has an even degree, its end behavior will be the same in both directions: as x approaches ±∞, f(x) approaches ±∞.

Front

What does it mean if a polynomial has an odd degree?

Back

If a polynomial has an odd degree, its end behavior will be opposite in each direction: as x approaches -∞, f(x) approaches -∞, and as x approaches +∞, f(x) approaches +∞.

Front

What is the significance of the leading coefficient?

Back

The leading coefficient of a polynomial affects the direction of the end behavior. If it is positive, the polynomial rises to the right; if negative, it falls to the right.

Front

How do you find the degree of the polynomial 9x^5 - x^4 + 2x?

Back

The degree of the polynomial 9x^5 - x^4 + 2x is 5, as the highest power of x is 5.

Front

What is the end behavior of the polynomial P(x) = -2x^4 + 3x^2 - 1?

Back

As x approaches ±∞, P(x) approaches -∞ because the leading term is negative and has an even degree.

Front

What is the degree of the polynomial function P(x) = 3x^4 - 7x^2 - 2x^7 - x + 4?

Back

The degree of the polynomial function P(x) is 7, as the highest power of x is 7.