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

What does it mean for a zero of a polynomial to 'cross' the x-axis?

Back

A zero crosses the x-axis if the polynomial changes sign at that zero, indicating an odd multiplicity.

Front

What does it mean for a zero of a polynomial to 'bounce' on the x-axis?

Back

A zero bounces on the x-axis if the polynomial does not change sign at that zero, indicating an even multiplicity.

Front

How do you determine the multiplicity of a zero?

Back

The multiplicity of a zero is determined by the exponent of the factor associated with that zero in the polynomial's factored form.

Front

What is the factored form of a polynomial?

Back

The factored form of a polynomial expresses it as a product of its linear factors, which can reveal its zeros.

Front

If a polynomial has a zero of multiplicity 3, what happens at that zero?

Back

If a polynomial has a zero of multiplicity 3, it crosses the x-axis at that zero.

Front

What is the factored form of f(x) = x^3 + 7x^2 - 2x - 14?

Back

f(x) = (x^2 - 2)(x + 7) is the factored form.

Front

What are the zeros of the polynomial f(x) = (x - 2)(x - 3)(x + 1)?

Back

The zeros are x = 2, x = 3, and x = -1.

Front

How do you factor the polynomial f(x) = x^4 - 9x^2 completely?

Back

f(x) = x^2(x + 3)(x - 3) is the complete factorization.

Front

What is the significance of the leading coefficient in determining end behavior?

Back

The leading coefficient determines whether the ends of the graph rise or fall as x approaches infinity or negative infinity.