Question

The polynomial \(3 x^{4}-2 x^{3}+13 x^{2}-5\) is of degree____ The leading coefficient is _____

Short answer

The polynomial degree is 4. The leading coefficient is 3.

Step-by-step solution

Identify the polynomial

Given polynomial is \[3 x^{4} - 2 x^{3} + 13 x^{2} - 5\]. This polynomial includes terms with different powers of \(x\).

Determine the degree

The degree of a polynomial is the highest power of the variable \(x\). In \[3 x^{4}-2 x^{3}+13 x^{2}-5\], the term with the highest power of \(x\) is \(3 x^{4}\). Therefore, the degree of the polynomial is \(4\).

Identify the leading coefficient

The leading coefficient is the coefficient of the term with the highest power. For the term \(3 x^{4}\), the coefficient is \(3\). Thus, the leading coefficient is \(3\).

Key concepts

Leading Coefficient

In a polynomial, the leading coefficient is a crucial element because it significantly impacts the polynomial's behavior for large values of the variable. The leading coefficient is defined as the coefficient of the term that has the highest power of the variable. For example, in the polynomial

3x^4 - 2x^3 + 13x^2 - 5

, the highest power of the variable x is 4, and the term corresponding to this highest power is

3x^4

. Therefore, the leading coefficient is 3.

To identify the leading coefficient, follow these simple steps:

• Find the term with the highest power in the polynomial.
• Look at the coefficient of that term - this is your leading coefficient.

It's vital to determine the leading coefficient correctly as it influences many aspects of the polynomial, such as end behavior and the leading term's dominance when the polynomial is plotted graphically.

Highest Power

The highest power in a polynomial—is also referred to as the degree of the polynomial. The degree gives us significant insight into the polynomial's structure and behavior. To find the highest power, examine the exponents of each term's variable.

Consider the polynomial:

3x^4 - 2x^3 + 13x^2 - 5

Each term has x raised to different powers: 4, 3, and 2, with the term

-5

having no x and thus implicitly representing

0

power. The highest exponent in this equation is 4: from the term

3x^4

. Therefore, the polynomial's degree is 4.

Understanding the degree is essential because:

• It tells us the maximum number of roots the polynomial can have.
• It determines the polynomial's end behavior.

The degree is fundamental when analyzing and solving polynomial equations.

Polynomial Terms

A polynomial consists of multiple terms, each term being a product of a coefficient and a variable raised to a power. These terms are usually separated by addition or subtraction signs. Let's break down the polynomial

3x^4 - 2x^3 + 13x^2 - 5

into its individual terms:

• 3x^4

  : This term has a coefficient of 3 and a variable raised to the power 4.

• -2x^3

  : Here, the coefficient is -2, and the variable is raised to the power 3.

• 13x^2

  : This term has a coefficient of 13 and the variable raised to the power 2.

• -5

  : This is a constant term, where the variable is raised to the power 0.

Each term plays a unique role in forming the overall behavior and shape of the polynomial when it's graphed.Recognizing each term and how they contribute individually helps in simplifying, adding, subtracting, and multiplying polynomials, thus making it easier to handle polynomial functions in various mathematical scenarios.