Without graphing, answer true or false to each statement....

Without graphing, answer true or false to each statement. Then, support your answer by graphing.
If a polynomial function of even degree has a negative leading coefficient and a positive $y$ -value for its $y$ -intercept, it must have at least two real zeros.

Key Concepts

Real Zeros

Real zeros of a polynomial function are the solutions to the equation f(x) = 0, where the graph crosses or touches the x-axis. By analyzing the end behavior and employing the Intermediate Value Theorem, one can determine the minimum number of real zeros without graphing, based on the changes in sign of the function between various intervals.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values of opposite sign on an interval, then it must cross zero at least once within that interval. Since polynomial functions are continuous everywhere, this theorem can be applied to deduce the existence of real zeros based on changes in sign.

Leading Coefficient and End Behavior

The leading coefficient of a polynomial function determines the direction in which the graph heads as x becomes large in magnitude. For an even degree polynomial, a negative leading coefficient means that both ends of the graph point downward. This characteristic, when combined with other factors like the y-intercept, helps predict the number of times the graph may cross the x-axis.

Even Degree Polynomials

Polynomials with an even degree have their end behavior such that as x approaches both positive and negative infinity, the function values will tend to the same direction (both upward or both downward). This property forms the basis for understanding where the function may cross the x-axis, as the behavior at the extremes is uniform.

y-intercept

The y-intercept of a function is the value of the function when x equals zero, indicating where the graph crosses the y-axis. A positive y-intercept means that at x = 0, the function’s value is positive, which can be an important clue when analyzing the transition of the function’s output from positive to negative values.

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