Question

What is the leading coefficient of $$ 5 x^4-6 x^2+2 x^6+1+7 x^3-x ? $$ A. 2 B. 4 C. 5 D. 7

Short answer

A. 2

Step-by-step solution

Identify the degrees of each term

Look at each term of the polynomial and identify its degree. Recall, the degree is the highest power of the variable in the term: - For the term \(5x^4\), the degree is 4. - For the term \(-6x^2\), the degree is 2. - For the term \(2x^6\), the degree is 6. - For the term \(7x^3\), the degree is 3. - For the term \(-x\), the degree is 1. - The constant term (1) has a degree of 0.

Determine the highest degree

Among the degrees identified, the highest degree is 6 associated with the term \(2x^6\).

Extract the leading coefficient

The leading coefficient is the coefficient of the term with the highest degree. In the term \(2x^6\), the coefficient is 2.

Select the correct answer

From the provided choices, identify the option that matches the leading coefficient. Thus, the answer is A. 2.

Key concepts

Polynomial Degree

A polynomial is an expression comprising terms, each formed by a constant multiplied by a variable raised to a non-negative integer power. For instance, in the polynomial \(5x^4 - 6x^2 + 2x^6 + 1 + 7x^3 - x \), each term has its own degree.
The degree of a term is the exponent of the variable in that term. For example, the term \(5x^4\) has a degree of 4, as indicated by the exponent.
When solving polynomial-related problems, like finding the leading coefficient, identifying the degree of each term is crucial. You can find the degree by checking the highest power of the variable, like in the term \(2x^6\) where the degree is 6.
The polynomial degree overall is the highest degree among all the terms in the polynomial. In our example, the highest degree is 6, making the polynomial a 6th-degree polynomial. Understanding this helps us proceed to identify the leading coefficient.

Coefficients in Polynomials

A coefficient is the numerical part of a term in a polynomial. It is the number multiplied by the variable(s) in the term. For instance:

• In the term \(5x^4\), the coefficient is 5.
• For \(-6x^2\), the coefficient is -6.
• In \(2x^6\), the coefficient is 2.

When solving problems, distinguishing between coefficients and variables is important. Confusing the two can lead to incorrect solutions. The coefficient provides the weight or impact of its associated variable term within the polynomial.
To solve for the leading coefficient, we look for the term with the highest degree per the variable’s power. In our example \(2x^6\), the coefficient is 2. Therefore, the leading coefficient is 2, a concept that is fundamental in polynomial analysis.

GED Math Problem Solving

Understanding polynomials is essential for solving math problems, especially in exams like the GED. The General Educational Development (GED) tests assess a wide variety of skills, including problem-solving with polynomials.
When solving a polynomial problem on the GED, here are useful steps to follow:

• Identify and list all terms in the polynomial.
• Determine the degree of each term by examining the exponent of the variable.
• Find the term with the highest degree, making it the leading term.
• Extract the coefficient of the leading term as the leading coefficient.

For example, consider the polynomial \(5x^4 - 6x^2 + 2x^6 + 1 + 7x^3 - x \). By following these steps, we recognize \(2x^6\) as the leading term. Its coefficient is 2, making it the leading coefficient.
Practicing these steps enhances problem-solving efficiency, ensuring that polynomial questions on the GED are approached logically and confidently.