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. It is determined by the leading term of the polynomial.

Front

Define the degree of a polynomial.

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression. It indicates the polynomial's growth rate and the number of roots it can have.

Front

How do you find the zeros of a polynomial?

Back

To find the zeros of a polynomial, set the polynomial equal to zero and solve for the variable. The solutions are the x-values where the graph intersects the x-axis.

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: if positive, the graph rises to the right; if negative, it falls to the right.

Front

Factor the polynomial: f(x) = x^4 - 9x^2.

Back

f(x) = x^2(x + 3)(x - 3).

Front

Describe the end behavior of the polynomial f(x) = x^4 - 5x^2 + 2.

Back

As x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞.

Front

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

Back

A polynomial of degree n can have up to n real zeros, counting multiplicities.

Front

Explain how to determine the end behavior of f(x) = 3x^4 + 2x - 5.

Back

As x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞ due to the leading term 3x^4.

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 irreducible polynomials over the real numbers.

Front

Identify the zeros of the polynomial f(x) = x^2 - 4.

Back

The zeros are x = 2 and x = -2.