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Chapter 4: Problem 3

Fill in each blank with the correct response. The degree of the term \(-3 x^{9}\) is _____.

Short Answer

Expert verified
9

Step by step solution

01

Identify the Term

Look at the given term \(-3 x^{9}\).
02

Locate the Exponent

Focus on the exponent in the term. In this case, the exponent is 9.
03

Determine the Degree

The degree of a term in a polynomial is determined by the highest exponent of the variable. Therefore, the degree of the term \(-3 x^{9}\) is 9.

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Key Concepts

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

Understanding Polynomials
A polynomial is a type of algebraic expression composed of variables, constants, and exponents. It's formed by adding or subtracting terms. Each term is a product of a constant (called the coefficient) and a variable raised to a non-negative integer exponent. For example, in the polynomial equation \(5x^3 + 2x^2 - x + 7\), there are four terms: \(5x^3\), \(2x^2\), \(-x\), and 7.

Polynomials are widely used in algebra, calculus, and various areas of mathematics. They help describe relationships and patterns, and they are foundational to understanding more complex mathematical concepts.

  • The constants in polynomials are known as coefficients.
  • The degree of a polynomial is determined by the term with the highest exponent.
  • Polynomials with one, two, or three terms are often called monomials, binomials, and trinomials, respectively.
Degree of a Polynomial Term
In algebra, the degree of a polynomial term is a critical concept. The degree of a term is determined solely by the exponent of its variable. It tells us about the highest power to which the variable is raised.

For example, let's look at the term \(-3x^9\). Here, the exponent is 9, so the degree of this term is 9. This means the variable \(x\) is raised to the 9th power.

Some key points to remember regarding the degree of a term:
  • If a term contains more than one variable, the degree is the sum of the exponents of all the variables in that term. For instance, in the term \(2x^3y^2\), the degree is \(3 + 2 = 5\).
  • For constant terms (terms with no variables), the degree is 0.
  • The degree of a polynomial is the highest degree of any of its terms.
Exponents in Algebra
Exponents play a crucial role in algebra, indicating how many times a number, known as the base, is multiplied by itself. In an expression like \(x^9\), the number 9 is the exponent, and \(x\) is the base.

When working with exponents, there are several important rules to remember:
  • Product of Powers Rule: When multiplying two powers that have the same base, you can add the exponents. For example, \(x^3 \times x^2 = x^{3+2} = x^5\).
  • Power of a Power Rule: When raising a power to another power, you multiply the exponents. For instance, \((x^3)^2 = x^{3 \times 2} = x^6\).
  • Power of a Product Rule: When raising a product to an exponent, you can raise each factor to the exponent. For example, \((xy)^3 = x^3 \times y^3\).
  • Quotient of Powers Rule: When dividing two powers with the same base, you subtract the exponents. For example, \(x^5 / x^2 = x^{5-2} = x^3\).
Exponents make it easier to write and work with very large or very small numbers, helping simplify complex algebraic expressions.

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