Question

Identify the degree of each polynomial. $$5+6 a+3 a^{3}+4 a^{2}-a^{5}$$

Short answer

Answer: The degree of the given polynomial is 5.

Step-by-step solution

Write the polynomial in descending order

First, we will rewrite the given polynomial in descending order of the powers of 'a': $$-a^{5} + 3a^{3} + 4a^{2} + 6a+ 5$$

Identify the highest power of 'a'

Now that the polynomial is in descending order, we can easily identify the highest power of the variable 'a'. The highest power is 5, as seen in the term \(-a^5\).

The degree of the polynomial

Since the highest power of 'a' in the polynomial is 5, the degree of the polynomial is 5.

Key concepts

Descending Order

When dealing with polynomials, arranging the terms in descending order is a crucial step that simplifies understanding and solving problems. Descending order means organizing the terms from the one with the highest power down to the lowest or constant term.

• In a polynomial, each term includes a variable raised to a power, called the exponent.
• By listing terms in descending order, you begin with the term that has the highest exponent.
• This systematic arrangement allows for easier identification of significant properties like the degree of the polynomial.

Let's take the polynomial from the initial exercise, which is given as:\[5+6a+3a^{3}+4a^{2}-a^{5}\]Rewritten in descending order, it becomes:\[-a^{5} + 3a^{3} + 4a^{2} + 6a +5\]This reorganization helps us immediately spot the term with the highest power, critical for identifying key characteristics of the polynomial.

Highest Power

The highest power in a polynomial is fundamentally important as it gives us valuable information about the polynomial's overall characteristics. The power refers to the exponent of the variable in a term.

• The highest power indicates the leading term, which significantly impacts the polynomial's behavior, especially for larger inputs.
• Identifying this term allows mathematicians to predict the polynomial's growth and trends.

In our rearranged polynomial \(-a^{5} + 3a^{3} + 4a^{2} + 6a + 5\), the term \(-a^{5}\) has the highest power, 5.By focusing on this highest power, we can draw conclusions about the polynomial's shape and anticipate its general form and degree.

Identifying Polynomial Degree

The degree of a polynomial is a critical attribute in mathematics, as it affects both its graphical representation and behavior. The degree is defined by the highest power of the variable present in the polynomial.

• A polynomial's degree provides insight into the number of roots or intercepts it might possess.
• It also suggests the maximum number of bends or turns the graph of the polynomial will exhibit.

In the example from the exercise, after arranging the polynomial \(-a^{5} + 3a^{3} + 4a^{2} + 6a + 5\), we identify the highest power 5, hence the degree of the polynomial is 5.This means the polynomial potentially has up to five roots, and its graph can have up to four turning points, highlighting the ease with which understanding the degree can simplify other relevant mathematical tasks.