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Chapter 5: Problem 12

Give the numerical coefficient and the degree of each term. $$ -y^{9} $$

Short Answer

Expert verified
The numerical coefficient is -1 and the degree is 9.

Step by step solution

01

Identify the term

The given term is ewline -y^{9}. We need to find the numerical coefficient and the degree of this term.
02

Find the numerical coefficient

Look at the constant number in front of the variable. In ewline -y^{9}, the number in front is -1. Therefore, the numerical coefficient is -1.
03

Determine the degree of the term

The degree of a term is the exponent of the variable. For ewline -y^{9}, the exponent is 9. Therefore, the degree is 9.

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Key Concepts

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

Numerical Coefficient
In algebra, every term consists of a constant number and one or more variables. The numerical coefficient is the constant number that multiplies the variable(s). For example, in the term \(\-y^{9}\), the numerical coefficient is \(-1\) because it is the constant that multiplies the variable \(y^{9}\). The coefficient gives us a sense of 'how much' of the variable we have. If a term does not visibly show a coefficient, like \(y\), it actually means the coefficient is \(1\).
Degree of a Term
The degree of a term refers to the exponent of the variable in that term. In our example term \(-y^{9}\), we look at the exponent attached to \(y\), which is \(9\). So, the degree of the term \(-y^{9}\) is \(9\).
Here are a few more examples to understand this concept better:
  • In \(3x^{2}\), the degree is \(2\).
  • In \(4a^{5}b^{3}\), the degree is the sum of the exponents, which is \(5+3=8\).
  • For a constant term like \(7\), the degree is \(0\) because there are no variables.
The degree helps us understand the complexity or 'power' of the term in the polynomial.
Polynomial Terms
A polynomial is a mathematical expression made up of terms that are added or subtracted from each other. Each term in a polynomial has its own coefficient and degree.
For example, consider the polynomial: \(3x^2 - 4x + 7\).
This polynomial has three terms:
  • \(3x^2\) with a coefficient \(3\) and a degree of \(2\),
  • \(-4x\) with a coefficient \(-4\) and a degree of \(1\),
  • \(7\) which is a constant term with a degree of \(0\).
When working with polynomials, it's vital to understand each term's coefficient and degree, as they play essential roles in polynomial operations like addition, subtraction, and multiplication.

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Most popular questions from this chapter

In Exercises 65鈥�74, the factors involve fractions or decimals. Apply the methods of this sec- tion, and find each product. $$ \left(4 x-\frac{2}{3}\right)\left(4 x+\frac{2}{3}\right) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 m^{3} x^{2}\right)^{-1}\left(3 m^{4} x\right)^{-3}}{\left(m^{2} x^{3}\right)^{3}\left(m^{2} x\right)^{-5}} $$

Write each number in scientific notation. $$ 0.875 $$

Find each product. $$ (3 r+2 s)\left(r^{3}+2 r^{2} s-r s^{2}+2 s^{3}\right) $$

Simplify each expression. $$ 7 x-(5+5 x)+3 $$
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