Writing Polynomial Functions

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression that involves 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 a_n ≠ 0.

Front

What are the zeros of a polynomial?

Back

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

Tags

CCSS.HSF-IF.C.7C

Front

How do you write a polynomial in standard form?

Back

A polynomial is in standard form when its terms are arranged in descending order of the degree of the variable. For example, f(x) = 2x^3 + 3x^2 - x + 5 is in standard form.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial f(x) = 4x^5 - 3x^2 + 2, the degree is 5.

Front

How do you find the factored form of a polynomial?

Back

To find the factored form of a polynomial, identify its zeros and express the polynomial as a product of its linear factors. For example, if the zeros are x = 1 and x = -2, the factored form is f(x) = (x - 1)(x + 2).

Tags

CCSS.HSF-IF.C.7C

Front

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

Back

Each zero of a polynomial corresponds to a factor of the form (x - zero). For example, if a polynomial has zeros at x = 3 and x = -1, its factors are (x - 3) and (x + 1).

Tags

CCSS.HSF-IF.C.7C

Front

How do you determine the number of zeros a polynomial can have?

Back

A polynomial of degree n can have at most n real zeros. However, it may have fewer real zeros and some complex zeros.

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. If it's positive, the graph rises to the right; if negative, it falls to the right.

Front

What is synthetic division?

Back

Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form (x - c). It is faster than long division and is used to find polynomial quotients and remainders.

Tags

CCSS.HSA.APR.D.6

Front

How do you graph a polynomial function?

Back

To graph a polynomial function, identify its zeros, determine the end behavior, and plot key points. Use the zeros to find where the graph crosses the x-axis and analyze the behavior between and beyond the zeros.