Writing Polynomial Functions

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 (or roots) of a polynomial function are the values of x for which the function equals zero. They can be found by factoring the polynomial or using the quadratic formula.

Front

How do you find the zeros of the function y=(x + 2)(x - 3)(3x - 5)?

Back

Set the function equal to zero: (x + 2)(x - 3)(3x - 5) = 0. The zeros are x = -2, x = 3, and x = 5/3.

Front

What is the relationship between the roots and the factors of a polynomial?

Back

Each root of a polynomial corresponds to a factor of the polynomial. If r is a root, then (x - r) is a factor.

Front

How do you write a polynomial given its roots?

Back

To write a polynomial from its roots, use the factored form: f(x) = a(x - r1)(x - r2)...(x - rn), where r1, r2, ..., rn are the roots and a is a non-zero constant.

Front

What is multiplicity in polynomial functions?

Back

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

Front

Given the roots 1, -4, and 1/2, how do you write the polynomial?

Back

The polynomial can be written as f(x) = a(x - 1)(x + 4)(x - 1/2). If a = 2, then f(x) = 2(x - 1)(x + 4)(x - 1/2).

Front

What is the standard form of a polynomial?

Back

The standard form of a polynomial is expressed as f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where the terms are arranged in descending order of the degree.

Front

How do you determine the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the expression. For example, in f(x) = 2x^3 + 3x^2 - x + 5, the degree is 3.

Front

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

Back

The leading coefficient is the coefficient of the term with the highest degree. It affects the end behavior of the polynomial graph.