Problem 31 - Algebra 1: Concepts and Skills

Question

Write the polynomial in standard form. Then identify the polynomial by degree and by the number of terms. $$ -4 b^{2}+7 b^{3} $$

Step-by-Step Solution

Answer

The polynomial in standard form is \(7 b^{3}-4 b^{2}\). The degree of the polynomial is 3 and it has 2 terms.

Step 1: Reorder the Polynomial

Reorder the polynomial in descending order of their degrees. This means that, the term with the highest power comes first. Given \(-4 b^{2}+7 b^{3}\), when we reorder we get \(7 b^{3}-4 b^{2}\).

Step 2: Identify the Degree of the Polynomial

The degree of the polynomial is the power of the variable of the term with the highest degree. From \(7 b^{3}-4 b^{2}\), the term with the highest power is \(7 b^{3}\). Therefore, the degree of the polynomial is 3.

Step 3: Count the Number of Terms

The number of terms in a polynomial is the number of parts separated by + or - signs. From \(7 b^{3}-4 b^{2}\), we can see that there are 2 terms in the polynomial as they are separated by the - sign.

Key Concepts

standard form

Writing a polynomial in standard form is all about organization. The goal is to rearrange the terms so that they are ordered from the highest degree to the lowest. This helps in easily recognizing the leading term, which is crucial for understanding the polynomial.

When you look at a polynomial, each term will have a coefficient (the number in front of the variable) and a degree (the exponent on the variable). By knowing this, you can:

• Identify the term with the highest degree and place it first.
• Order the remaining terms in decreasing order of their degrees.
• Ensure no terms are left in disorder by degrees.

This systematized approach adds clarity to polynomials. For our given polynomial \(-4 b^{2}+7 b^{3}\), we rearrange it to become \(7 b^{3}-4 b^{2}\). Now, it is in the standard form with the highest degree term \(7 b^3\) clearly leading.

degree of a polynomial

Understanding the degree of a polynomial is key in determining many of its properties. The degree is essentially the highest exponent of a variable in the polynomial.

To find the degree in our polynomial \(7 b^{3}-4 b^{2}\), observe:

• Identify the term with the biggest exponent, which is the leading term.
• Here it's \(7 b^{3}\), and the exponent is 3.
• Thus, the degree of the polynomial is 3.

Recognizing the degree allows you to classify the polynomial and predict some of its behavior in equations or graphs. For instance, a degree of 3 often hints at a cubic graph, giving you insight into potential solutions and the shape of the graph.

number of terms

Counting the number of terms in a polynomial is straightforward but vital. A "term" in a polynomial is each distinct part surrounded by + or - signs. Each term includes a coefficient and a variable raised to an exponent.

Let’s dissect the polynomial \(7 b^{3}-4 b^{2}\):

• Each separated by a minus or plus sign represents an individual term.
• The given polynomial has two distinct parts: \(7 b^{3}\) and \(-4 b^{2}\).
• So, in this case, there are 2 terms.

This tells us that the given polynomial is a "binomial," which means it has exactly two terms. Knowing the number of terms helps in classifying the polynomial and in understanding the simplification processes it may undergo during algebraic operations.