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Chapter 4: Problem 88

Give the degree of each polynomial. $$-25$$

Short Answer

Expert verified
The degree of the polynomial '-25' is 0.

Step by step solution

01

Indentify the Polynomial

The first step is to recognize the type of the expression given. Here, -25 is a constant.
02

Find the Degree of the Polynomial

Next, understand that the degree of a polynomial is defined as the highest power of its variable. However, a constant does not contain a variable, so we can consider the constant -25 as a polynomial of degree zero. Therefore, the degree of the polynomial '-25' is 0.

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Key Concepts

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

Constant Polynomial
A constant polynomial is a special type of polynomial that does not contain any variables. In simpler terms, it means that a constant polynomial is just a number by itself, like 5, -3, or even 0. These numbers stand alone without an accompanying variable like \(x\) or \(y\).

The uniqueness of constant polynomials lies in their simplicity and the fact that they always evaluate to the same value, regardless of any substitution.
  • A constant polynomial can be thought of as a horizontal line on a graph, representing a constant value.
  • These types of polynomials are the most basic form. Every number is technically a polynomial on its own.
For example, the expression \(-25\) is a constant polynomial, showcasing a single numeric value without any variable. This makes constant polynomials an interesting study in mathematics for their predictable nature.
Degree of Zero
The concept of degree is very important in polynomials. When it comes to a constant polynomial, for instance, an expression like \(-25\), we have to delve into what 'degree' means. The degree of a polynomial is determined by finding the highest power of its variable.

Since a constant polynomial has no variable, it's natural to wonder what its degree is.
  • The degree of any constant polynomial is zero.
  • This happens because we think of a constant like \(-25\) as \(-25 x^0\). Since the power of \(x\) is 0 here, the polynomial's degree is 0.
This zero-degree property distinguishes constant polynomials from others, emphasizing their nature devoid of any variables or exponents.
Identifying Polynomials
Polynomials can be identified based on their terms, coefficients, and exponents. Identifying a polynomial involves recognizing its type and its degree which tells us about the polynomial's complexity and behavior.

When you come across an expression, the first step is to check if it includes variables raised to a power.
  • If it does not have a variable, it's a constant polynomial.
  • If it has a variable with an exponent, then it has a certain degree based on the highest power.
  • For instance, the expression \(3x^2 - 4x + 7\) is a polynomial with a degree of 2 because of the \(x^2\) term.
Knowing how to identify a polynomial helps in classifying and understanding its properties, essential for any further mathematical operations.

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Most popular questions from this chapter

The radius of one millstone in the illustration is 3 meters greater than the radius of the other, and their areas differ by \(15 \pi\) square meters. Find the radius of the larger millstone.

Perform the operations. $$\begin{aligned}\\\ &-3 x^{3} y^{6}+2\left(x y^{2}\right)^{3}-(3 x)^{3} y^{6} \end{aligned}$$

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The revenue \(r\) (in dollars) that a manufacturer of desk chairs receives is given by the polynomial function $$r=f(d)=-0.08 d^{2}+100 d$$ where \(d\) is the number of chairs manufactured. Find the revenue received when 815 chairs are manufactured.
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