Zeros

Front

What is the end behavior of a polynomial function as x approaches infinity?

Back

As x approaches infinity, the end behavior of the graph can vary depending on the leading coefficient and degree of the polynomial. For even-degree polynomials with a positive leading coefficient, y approaches infinity. For odd-degree polynomials with a positive leading coefficient, y approaches infinity as x approaches infinity and y approaches negative infinity as x approaches negative infinity.

Front

What is the end behavior of a polynomial function as x approaches negative infinity?

Back

As x approaches negative infinity, the end behavior of the graph can vary depending on the leading coefficient and degree of the polynomial. For even-degree polynomials with a positive leading coefficient, y approaches infinity. For odd-degree polynomials with a positive leading coefficient, y approaches negative infinity as x approaches negative infinity.

Front

What is factored form of a polynomial?

Back

The factored form of a polynomial is an expression that represents the polynomial as a product of its linear factors. For example, f(x) = (x - r1)(x - r2)...(x - rn), where r1, r2, ..., rn are the roots or zeros of the polynomial.

Front

How do you find the zeros of a polynomial function?

Back

To find the zeros of a polynomial function, set the function equal to zero and solve for x. This can involve factoring the polynomial, using the quadratic formula, or applying synthetic division.

Front

What does the multiplicity of a zero indicate?

Back

The multiplicity of a zero indicates how many times a particular zero appears as a root of the polynomial. A zero with an odd multiplicity will cross the x-axis, while a zero with an even multiplicity will touch the x-axis and turn around.

Front

What is the significance of the leading coefficient in a polynomial function?

Back

The leading coefficient determines the direction of the end behavior of the polynomial function. If the leading coefficient is positive, the graph will rise to the right; if it is negative, the graph will fall to the right.

Front

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

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

How can you determine the number of turning points in a polynomial graph?

Back

The maximum number of turning points in a polynomial graph is one less than the degree of the polynomial. For example, a polynomial of degree 4 can have up to 3 turning points.

Front

What is the role of the x-intercepts in the graph of a polynomial function?

Back

The x-intercepts of a polynomial function are the points where the graph crosses or touches the x-axis. These points correspond to the zeros of the function.

Front

What is the difference between a zero and an x-intercept?

Back

A zero of a polynomial function is a value of x that makes the function equal to zero, while an x-intercept is the point on the graph where the function crosses the x-axis. They are often the same, but the term 'zero' refers to the value, while 'x-intercept' refers to the point.