Question

In Exercises 7-12, determine whether the algebraic expression is a polynomial. If it is, write the polynomial in standard form and state its degree.\(2 x-3 x^{3}+8\)

Short answer

The given algebraic expression \(-3x^{3} + 2x + 8\) is indeed a polynomial. In standard form, it is written as \(-3x^{3} + 2x + 8\), and it has a degree of 3.

Step-by-step solution

Identify the Algebraic Expression

Given the algebraic expression, \(2x - 3x^{3} + 8\). Determine if it is a polynomial.

Check if Expression is a Polynomial

A polynomial should only contain terms with whole number exponents. Looking at the expression \(2x - 3x^{3} + 8\), all its terms \(2x\), \(-3x^{3}\) and \(8\) are completely simplified terms and the powers of the variable x are whole numbers (1 and 3). Hence the expression is a polynomial.

Arrange Polynomial in Standard Form

In standard form, a polynomial is organized with powers of the variable in decreasing order. Therefore, \(2x - 3x^{3} + 8\) becomes \(-3x^{3} + 2x + 8\).

State Degree of Polynomial

The degree of a polynomial is given by the highest power of the variable. Looking at the polynomial, the highest power of x is 3. Hence, the degree of the polynomial is 3.

Key concepts

Polynomial

A polynomial is a type of mathematical expression that plays a fundamental role in algebra and represents a sum of terms. Each term is a product of a constant coefficient and a variable raised to a non-negative integer exponent.

A key characteristic of a polynomial is its components: variables, exponents, and coefficients. Variables are symbols usually represented by letters like 'x' or 'y' that can take on different values. Exponents indicate how many times to multiply the variable by itself, while coefficients are the numbers multiplying the variables.

For example, in the expression \(2x - 3x^{3} + 8\), \(2x\) and \(3x^{3}\) are two of the terms where 'x' is the variable. Here, the numbers 2 and 3 are coefficients, and 1 and 3 are their respective exponents.

Standard Form of Polynomial

When we arrange a polynomial, we prefer to write it in its standard form. This form provides a clear and organized way to understand the structure of the polynomial.

The standard form of a polynomial involves ordering the terms from the highest exponent to the lowest exponent of a given variable, which typically is 'x'. To visualize this, consider the expression from our example, \(2x - 3x^{3} + 8\). Arranged in standard form, it becomes \( -3x^{3} + 2x + 8 \), which neatly aligns the terms from the highest power of 'x' to the constant term.

Why Use Standard Form?

Using the standard form makes it easier to compare polynomials and perform operations like addition, subtraction, and multiplication. It provides a uniform approach to working with these expressions, which is particularly useful when solving algebraic problems.

Polynomial Degree

The degree of a polynomial is a fundamental concept that determines many of its properties. The degree is defined as the highest power of the variable that appears in the polynomial when it is written in standard form.

Returning to our example with the expression \( -3x^{3} + 2x + 8 \), we identify the term with the greatest exponent, which is \( -3x^{3} \). The exponent here is 3; hence, the polynomial's degree is 3. This indicates that it is a cubic polynomial.

The degree of a polynomial gives us a quick insight into the behavior of its graph. For instance, a polynomial of degree 1 (linear) will always graph as a straight line, while a polynomial of degree 3 can have curves and possibly more than one local maximum or minimum point.