Problem 15 - Algebra Form and Function

Question

Give the leading term. $$ 2 x^{7}-4 x^{11}+6 $$

Step-by-Step Solution

Answer

Answer: The leading term is \(-4x^{11}\).

Step 1: Identify the terms

The polynomial is given as: $$ 2x^{7} - 4x^{11} + 6 $$ The terms in this polynomial are: 1. \(2x^7\) 2. \(-4x^{11}\) 3. \(6\) Step 2: Determine the degree of each term

Step 2: Find the degree

The degree of a term is the exponent of the variable in that term. In this case: 1. The degree of the term \(2x^7\) is 7. 2. The degree of the term \(-4x^{11}\) is 11. 3. The degree of the constant term (6) is 0. Step 3: Identify the leading term

Step 3: Find the leading term

Among the terms, the term with the highest degree is the leading term. In this case, the term with the highest degree is \(-4x^{11}\), as 11 is the highest degree. Hence, the leading term is: $$ -4x^{11} $$

Key Concepts

Polynomial Degree

The degree of a polynomial is a critical concept that tells us about the polynomial's highest power of the variable. Understanding the degree helps in identifying the behavior of the polynomial when plotted on a graph, and it's crucial in problems related to comparing or arranging polynomials. The degree is determined by looking at the exponent of the variable in each of the terms within the polynomial.

The degree of each term is simply the largest exponent on the variable within that term. If a term has no variable, like a constant, its degree is zero. If you have a polynomial like the one given, \(2x^7 - 4x^{11} + 6\), then it includes terms with degrees of 7 (for \(2x^7\)), 11 (for \(-4x^{11}\)), and 0 (for 6).

To find the degree of the polynomial as a whole, you simply identify the term with the largest degree. In this case, \(-4x^{11}\) carries the highest exponent (11), making the degree of the polynomial 11.

Polynomial Terms

Understanding polynomial terms is fundamental when working with polynomials. Each separate part of a polynomial that involves the variable and its coefficients is a term. In our example polynomial, \(2x^7 - 4x^{11} + 6\), it has three distinct terms.

Listed out within the polynomial, terms can be:

• \(2x^7\): This term includes a coefficient of 2 and an exponent of 7.
• \(-4x^{11}\): This term has a coefficient of -4 and an exponent of 11.
• \(6\): This last term is a constant with no variable, but is still an important part of the polynomial.

Recognizing terms, their coefficients, and exponents is key to simplifying, adding, or subtracting polynomials. Each term is separately calculated based on the rules of arithmetic and algebra, without impacting the other terms unless specifically combined through these operations.

Exponent Rules

Exponent rules are essential tools in algebra that simplify expressions involving powers. In polynomials, exponent rules mainly help us simplify, differentiate, or integrate terms. When working through a polynomial problem, such as finding the leading term, exponent rules play an important role in determining the term with the highest degree.

Let's go over a few basic exponent rules:

• Product of Powers: When multiplying two terms with the same base, add the exponents (e.g., \(x^a \times x^b = x^{a+b}\)).
• Power of a Power: When raising a power to another power, multiply the exponents (e.g., \((x^a)^b = x^{a \times b}\)).
• Power of a Product: When a product is raised to a power, each factor in the product is raised to the power (e.g., \((xy)^a = x^a \times y^a\)).

These rules allow you to work with polynomials more confidently by enabling you to simplify and manipulate the expressions as needed. In the context of our example polynomial, these rules help in understanding why \(-4x^{11}\) is the leading term, due to its highest exponent.