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Chapter 5: Problem 12

Determine whether each expression is a polynomial. \(-10\)

Short Answer

Expert verified
-10 is a polynomial of degree zero.

Step by step solution

01

Understand the definition of a polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is given by: a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0where the coefficients (a_n, a_(n-1),..., a_0) are real numbers, and the exponents are whole numbers (non-negative integers).
02

Analyze the given expression

The given expression is -10Notice that this expression does not have a variable (like x) present; it is a constant.
03

Identify the degree of the expression

In polynomial terms, a constant like -10can be seen as -10x^0where the power of x is zero. Since any number raised to the power of zero is 1.
04

Conclusion

Since a constant can be considered a polynomial of degree zero, -10is a polynomial.

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Key Concepts

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

polynomial definition
Polynomials are fundamental elements in algebra. They consist of coefficients and variables raised to whole number powers. The general form of a polynomial in a single variable is written as: \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\)
  • Here, \(a_n, a_{n-1}, \ldots, a_1, a_0\) are coefficients.
  • The exponents \(n, n-1, \ldots, 1, 0\) are non-negative integers.
Polynomials can have one variable (like \(x\)) or multiple variables (like \(x\) and \(y\)). Polynomials are important because they can describe a wide range of mathematical relationships and models.
degree of polynomial
The degree of a polynomial is the highest power of the variable in the polynomial expression. It tells us the most prominent term in the polynomial.
  • For example, in \(4x^3 + 3x^2 + 2x + 1\), the degree is 3 because the highest exponent of \(x\) is 3.
  • In the polynomial \(-5 + 2x^2 + 3x^7\), the degree is 7.
  • A polynomial like \(6\) has a degree of 0 because it can be written as \(6x^0\).
Understanding the degree of a polynomial helps in identifying its behavior and graph properties. High degree polynomials can have complex graphs with many twists and turns.
constant term
A constant term in a polynomial is a term that does not contain any variable. It is simply a fixed number. For instance:
  • In the polynomial \(4x^3 + 3x^2 + 2x + 1\), the constant term is 1.
  • In \(-5 + 2x^2 + 3x^7\), the constant term is \(-5\).
Even though the constant term doesn't change with the variable, it plays an important role in the polynomial. It dictates the value of the polynomial when all the variables are zero. So, in the context of the example exercise, \(-10\) is a constant term and since it can be considered as \(-10x^0\), it is indeed a polynomial with a degree of zero.

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Most popular questions from this chapter

Simplify. $$ \left[\left(5^{-3}\right)^{2}\right]^{-1} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=5 x-1\\\ &g(x)=2 x^{2} \end{aligned} $$

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 4^{2} \square 4^{3} $$

To prepare for Section 5.3, review combining like terms and evaluating expressions (Sections 1.6 and 1.8). Combine like terms. $$ 5 a-7 b-8 a+b $$

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 4^{3} \square 3^{4} $$
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