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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where \(a_n \neq 0\) and \(n\) is a non-negative integer.

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. For example, in the polynomial $$y = -4x^5 + 3x^2 - 8x + 1$$, the degree is 5.

Front

How do you find the roots of a polynomial?

Back

The roots of a polynomial are the values of \(x\) that make the polynomial equal to zero. They can be found by factoring the polynomial or using the quadratic formula for polynomials of degree 2.

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, the polynomial with roots \(x = -2, x = 5, x = 3\) can be factored as $$y = (x + 2)(x - 5)(x - 3)$$.

Tags

CCSS.HSA.APR.B.3

Front

What does multiplicity mean in the context of polynomial roots?

Back

Multiplicity refers to the number of times a particular root appears in a polynomial. For example, in the polynomial $$y = (x + 2)(x - 7)^3$$, the root \(-2\) has a multiplicity of 1, and \(7\) has a multiplicity of 3.

Tags

CCSS.HSF-IF.C.7C

Front

How do you determine the degree of a polynomial from its graph?

Back

The degree of a polynomial can be determined from its graph by observing the number of turning points. A polynomial of degree \(n\) can have at most \(n-1\) turning points.

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. For example, in the polynomial $$y = 3x^4 + 2x^3 - x + 5$$, the leading coefficient is 3.

Front

What is the standard form of a polynomial?

Back

The standard form of a polynomial is when the terms are arranged in descending order of their degrees. For example, $$y = 2x^3 + 3x^2 - 5x + 1$$ is in standard form.

Front

What is the difference between a monomial, binomial, and trinomial?

Back

A monomial is a polynomial with one term (e.g., $$3x^2$$), a binomial has two terms (e.g., $$x^2 + 4$$), and a trinomial has three terms (e.g., $$x^2 + 4x + 4$$).

Front

How do you add and subtract polynomials?

Back

To add or subtract polynomials, combine like terms. For example, $$ (2x^2 + 3x) + (4x^2 - 2x) = 6x^2 + x $$.

Tags

CCSS.HSA.APR.A.1