Polynomial Functions and Their Zeros

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 are the zeros of a polynomial function?

Back

The zeros of a polynomial function are the values of x for which the function f(x) = 0. They are also known as the roots of the polynomial.

Front

How do you find the zeros of the polynomial y = x(x - 6)(x + 5)?

Back

Set each factor equal to zero: x = 0, x - 6 = 0 (x = 6), x + 5 = 0 (x = -5). Thus, the zeros are x = 0, x = 6, and x = -5.

Front

What does the degree of a polynomial represent?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression. It indicates the maximum number of zeros the polynomial can have.

Front

What is the end behavior of a polynomial function?

Back

The end behavior describes how the values of f(x) behave as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

Front

What is multiplicity in relation to polynomial zeros?

Back

Multiplicity refers to the number of times a particular zero is repeated in a polynomial. A zero with an even multiplicity will touch the x-axis, while a zero with an odd multiplicity will cross the x-axis.

Front

How do you determine the end behavior of the polynomial function f(x) = x^5?

Back

As x approaches -∞, f(x) approaches -∞; as x approaches +∞, f(x) approaches +∞. This is because the leading term is positive and has an odd degree.

Front

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

Back

The leading coefficient determines the direction of the graph as x approaches positive or negative infinity. A positive leading coefficient with an odd degree will rise to the right and fall to the left.

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. For example, y = (x - r1)(x - r2)...(x - rn) where r1, r2, ..., rn are the zeros.

Front

How do you find the degree of the polynomial y = (3x + 2)(x - 7)^3?

Back

Expand the polynomial: the degree of (3x + 2) is 1 and the degree of (x - 7)^3 is 3. The total degree is 1 + 3 = 4.