Problem 83 - Algebra 2

Question

Write each polynomial in standard form. Then classify it by degree and by number of terms. $$ -3 x^{2}+6-x^{3} $$

Step-by-Step Solution

Answer

Standard form: \( -x^{3} - 3x^{2} + 6 \). Degree: 3 (cubic). Number of terms: 3 (trinomial).

Step 1: Writing the polynomial in standard form

Reorder the polynomial so that the terms are in descending order of degree. That means, the term with the highest exponent is written first.

Step 2: Classify by degree

Determine the degree of the polynomial, which is the highest power of the variable x. The degree will be the exponent of the leading term after it has been rewritten in standard form.

Step 3: Classify by number of terms

Count the number of terms in the polynomial to classify it. A monomial has one term, a binomial has two terms, a trinomial has three terms, and a polynomial with more than three terms does not have a special name based on the number of terms.

Key Concepts

Polynomial Degrees

One of the fundamental aspects of understanding polynomials is knowing about polynomial degrees. The degree of a polynomial tells us the highest power of the variable within that expression. For instance, in the polynomial \( -3x^2 + 6 - x^3 \), we need to identify the highest exponent, which in this case is 3. Hence, this polynomial is said to have a degree of 3. This is crucial because the degree of a polynomial determines the behavior of its graph and the number of possible roots it may have.

With higher degrees, there can be more complexity in the polynomial's curve when graphed. For example, a linear polynomial (degree 1) will always graph to a straight line, a quadratic polynomial (degree 2) to a parabola, and higher degrees can result in more complex graphs with various turning points.

Ordering Polynomials

Ordering polynomials in standard form is a necessary skill for simplifying and solving them effectively. The standard form requires that we write the terms of a polynomial in descending order of their degrees, starting with the term containing the highest power. For the exercise polynomial \( -3x^2 + 6 - x^3 \), reordering gives us \( -x^3 -3x^2 + 6 \).

This reordering makes it easier to identify properties such as the leading term and leading coefficient, which are important in understanding the behavior of the polynomial function. It also ensures a consistent format for comparison between different polynomials and simplifies operations such as addition and subtraction of polynomials. Hence, this step is not just a matter of convention but of practical importance when working with polynomials.

Classifying Polynomials by Terms

Polynomials can also be classified based on the number of terms they have. This helps in understanding their structure and provides a simplified way to refer to them. The key classifications are:

• A monomial has one term, such as \(5x^3\).
• A binomial has two terms, like \(x^2 + 4\).
• A trinomial has three terms, exemplified by \(x^3 - 2x + 1\).

If a polynomial has more than three terms, it's generally just referred to as a polynomial without a specific name related to the number of terms. In our exercise, the polynomial \( -x^3 -3x^2 + 6 \) is classified as a trinomial since it contains three terms. Classifying by number of terms not only aids in communication but also in identifying specific forms and patterns, such as the difference of squares often found in binomials or the perfect square trinomials.