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Chapter 13: Problem 4

Find the degree of each polynomial. $$4 b^{2}$$

Short Answer

Expert verified
The degree of the polynomial is 2.

Step by step solution

01

Understand the components of the polynomial

A polynomial consists of terms which are made up of coefficients and variables with exponents. In this exercise, we identify the terms present in the polynomial.
02

Identify the degree of each term

The degree of a term in a polynomial is determined by the exponent of the variable. In the term \(4b^{2}\), the exponent of \(b\) is 2.
03

Determine the degree of the polynomial

The degree of a polynomial is the highest degree of its terms. Since \(4b^{2}\) is the only term, the degree of the polynomial is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients in Polynomials
The coefficient in a polynomial is a number that multiplies the variable. When you look at a term like \(4b^2\), the coefficient is the number 4. The role of the coefficient is crucial as it indicates the magnitude by which the variable is multiplied.
To identify a coefficient, just isolate the number in front of the variable(s).
  • If there's no number, usually it's an implied 1.
  • Negative signs are also part of the coefficient. For example, in \(-3x\), the coefficient is -3.

Understanding coefficients is fundamental because they affect the scale of the term they accompany. They do not, however, influence the degree of the polynomial.
Exponents in Polynomials
Exponents in polynomials are the powers to which variables are raised. They tell you how many times to multiply the variable by itself. For example, in \(b^2\), the variable \(b\) is multiplied by itself once, resulting in \(b \times b\).
Exponents determine the degree of the term, which is the highest exponent of the variable in that term.
  • The polynomial degree helps determine the function's behavior and complexity.
  • An exponent of 1 in a term, like \(b^1\), is typically written as just \(b\).
  • Zero exponents lead to the value being 1, like \(x^0 = 1\), but only if \(x\) is non-zero.

Remember, the higher the exponent, the more rapidly the term will grow as the variable increases in value.
Terms in Polynomials
A polynomial is made up of one or more terms combined through addition or subtraction. Each term consists of a coefficient, a variable, and an exponent. Understanding terms is essential when dealing with polynomials.
For instance, \(4b^2\) is a single term. If combined with another term, like \(3b\), it forms a polynomial: \(4b^2 + 3b\).
  • Terms are separated by plus or minus signs.
  • Each term can have different coefficients and exponents.
  • The degree of a polynomial is the largest exponent among all its terms.

By working with terms, you can organize and solve problems involving polynomials effectively. Knowing how to identify and work with the individual components of a polynomial term is key to mastering polynomial expressions.

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Most popular questions from this chapter

Determine whether each expression is a polynomial. Explain your reasoning. If it is, classify it as a monomial, binomial, or trinomial. $$8-\frac{2}{k}$$

Determine whether each equation or table represents a linear or nonlinear function. Explain. $$\begin{array}{|c|c|} \hline x & y \\ \hline-10 & 20 \\ \hline-9 & 18 \\ \hline-8 & 16 \\ \hline-7 & 14 \\ \hline \end{array}$$

Graph each function. $$y=x^{2}$$

Find sum. \(\left(2 x^{2}+5 x\right)+(9-7 x)\)

Graph each pair of equations on the same coordinate plane. Describe their similarities and differences. $$\begin{aligned} &y=x^{2}\\\ &y=3 x^{2} \end{aligned}$$
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