End Behavior & Zeros of Polynomials

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.

Front

How do you determine the end behavior of a polynomial function?

Back

To determine the end behavior, look at the leading term of the polynomial. The sign and degree of the leading term dictate the behavior as x approaches ±∞.

Front

What does it mean if a polynomial has an odd degree and a positive leading coefficient?

Back

If a polynomial has an odd degree and a positive leading coefficient, the end behavior will be: as x → -∞, f(x) → -∞ and as x → +∞, f(x) → +∞.

Front

What does it mean if a polynomial has an odd degree and a negative leading coefficient?

Back

If a polynomial has an odd degree and a negative leading coefficient, the end behavior will be: as x → -∞, f(x) → +∞ and as x → +∞, f(x) → -∞.

Front

What does it mean if a polynomial has an even degree and a positive leading coefficient?

Back

If a polynomial has an even degree and a positive leading coefficient, the end behavior will be: as x → -∞, f(x) → +∞ and as x → +∞, f(x) → +∞.

Front

What does it mean if a polynomial has an even degree and a negative leading coefficient?

Back

If a polynomial has an even degree and a negative leading coefficient, the end behavior will be: as x → -∞, f(x) → -∞ and as x → +∞, f(x) → -∞.

Front

What are the zeros of a polynomial function?

Back

The zeros of a polynomial function are the values of x for which f(x) = 0. They are the x-intercepts of the graph.

Front

How do you find the zeros of a polynomial function?

Back

To find the zeros, set the polynomial equal to zero and solve for x. This may involve factoring, using 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 indicates the maximum number of zeros it can have. A polynomial of degree n can have up to n real zeros.

Front

What is the significance of the multiplicity of a zero?

Back

The multiplicity of a zero indicates how many times that zero is repeated. A zero with an odd multiplicity will cross the x-axis, while one with an even multiplicity will touch the x-axis and turn around.