Polynomial from the graph

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer.

Front

What determines the end behavior of a polynomial function?

Back

The end behavior of a polynomial function is determined by the degree of the polynomial and the sign of the leading coefficient.

Front

What is the leading coefficient of a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It influences the direction of the graph as x approaches positive or negative infinity.

Front

How do you find the x-intercepts of a polynomial function?

Back

To find the x-intercepts, set the polynomial equal to zero and solve for x. The solutions are the x-intercepts.

Front

What does it mean if a polynomial touches the x-axis at an x-intercept?

Back

If a polynomial touches the x-axis at an x-intercept, it means that the corresponding factor has an even multiplicity, indicating that the graph does not cross the x-axis at that point.

Front

What does it mean if a polynomial crosses the x-axis at an x-intercept?

Back

If a polynomial crosses the x-axis at an x-intercept, it means that the corresponding factor has an odd multiplicity, indicating that the graph changes direction at that point.

Front

What is 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 maximum number of x-intercepts and the end behavior.

Front

How do you determine the end behavior of the polynomial f(x) = -2x^3 + 4?

Back

As x approaches -∞, f(x) approaches -∞; as x approaches +∞, f(x) approaches -∞. This is because the leading coefficient is negative and the degree is odd.

Front

What is the significance of the degree being even or odd in a polynomial?

Back

If the degree is even, the ends of the graph will point in the same direction. If the degree is odd, the ends of the graph will point in opposite directions.

Front

What is the result of multiplying (x + 2)(x - 3)?

Back

The result is x^2 - x - 6.