graphing polynomials

Front

What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (or indeterminates) raised to non-negative integer powers and coefficients. For example, f(x) = 2x^3 - 4x^2 + 3 is a polynomial.

Front

What does it mean for a polynomial to be increasing on an interval?

Back

A polynomial is increasing on an interval if, for any two points x1 and x2 in that interval where x1 < x2, the value of the polynomial at x1 is less than the value at x2 (f(x1) < f(x2)).

Front

What are the zeros of a polynomial?

Back

The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

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 x. This may involve factoring, using the quadratic formula, or other algebraic methods.

Front

What is a leading coefficient?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It determines the end behavior of the polynomial's graph.

Front

What does the degree of a polynomial indicate?

Back

The degree of a polynomial indicates the highest power of the variable in the polynomial. It affects the shape and number of turns in the graph.

Front

What is a y-intercept?

Back

The y-intercept of a polynomial is the point where the graph intersects the y-axis. It can be found by evaluating the polynomial at x = 0.

Front

How do you determine the y-intercept of f(x) = x^3 - 5x^2 - 25x + 125?

Back

To find the y-intercept, substitute x = 0 into the polynomial: f(0) = 0^3 - 5(0^2) - 25(0) + 125 = 125. Thus, the y-intercept is (0, 125).

Front

What is the significance of the sign of the leading coefficient?

Back

The sign of the leading coefficient determines the direction of the ends of the polynomial's graph: if positive, the graph rises to the right; if negative, it falls to the right.

Front

What does it mean for a polynomial to have complex roots?

Back

Complex roots occur in conjugate pairs and indicate that the polynomial does not intersect the x-axis at those points. They arise when the discriminant of a quadratic is negative.