Problem 50 - Algebra A Combined Function

Question

Identify the degrees of the terms and the degree of the polynomial. See Example 12. $$ 2 a^{2} b+10 a^{4} b-9 a b+6 $$

Step-by-Step Solution

Answer

Terms have degrees 3, 5, 2, and 0; polynomial degree is 5.

Step 1: Identify Degrees of Individual Terms

The first term is \(2a^2b\). Its degree is found by adding the exponents of \(a\) and \(b\): \(2 + 1 = 3\).The second term is \(10a^4b\). Its degree is \(4 + 1 = 5\).The third term is \(-9ab\). Its degree is \(1 + 1 = 2\).The fourth term is a constant \(6\), with a degree of \(0\).

Step 2: Identify the Degree of the Polynomial

The degree of a polynomial is the highest degree among its terms. From Step 1, the degrees of the terms are \(3, 5, 2,\) and \(0\). Therefore, the degree of the polynomial is \(5\), which is the highest degree of the given terms.

Key Concepts

Terms of a Polynomial

A polynomial is a mathematical expression composed of variables, coefficients, and constants combined through addition and subtraction. Each component of a polynomial separated by a plus or minus sign is known as a 'term'. Terms can feature variables raised to exponents and each exponent defines the nature of that term.
For example, in the polynomial:

• \(2a^2b\) is the first term where \(a^2b\) stands for the variables along with their exponents and 2 is the coefficient.
• \(10a^4b\) is the second term.
• \(-9ab\) is the third term.
• 6 is a simple constant term with no variables attached.

Understanding these constituents sets the foundation for determining other properties such as the degree of each term and ultimately, the degree of the whole polynomial.

Degree of a Term

The 'degree of a term' in a polynomial is crucial to understanding the term’s significance within the overall expression. It is defined by adding the exponents of all variables present in the term.

To illustrate this:

• The term \(2a^2b\) has 'a' raised to the 2nd power and 'b' to the 1st (since \(b = b^1\)), amounting to a degree of \(2 + 1 = 3\).
• For \(10a^4b\), you add the exponent 4 for 'a' to 1 for 'b', resulting in a degree of 5.
• The term \(-9ab\) has both \(a\) and \(b\) raised to the power of 1, thus 1 + 1 = 2 describes its degree.

Each term's degree is a critical measure that helps in further calculations like identifying the polynomial degree.

Constant Term

A constant term in a polynomial is the term that has no variables, meaning it is not multiplied by any variable and is defined purely as a numerical value. In the context of polynomial degree, the constant term is unique because its degree is always zero. This is due to the absence of variables with exponents in the term.

• In our exercise, the constant term is 6. As it stands alone without any variable components, its degree is \(0\).

Constant terms might seem simple, but they play a vital role, especially in determining roots and evaluating the polynomial at given variable values.

Identifying Polynomial Degree

Determining the degree of a polynomial involves finding the term with the highest degree. This gives insight into the behavior and scope of the polynomial.

In the exercise provided, once you've calculated the degrees of individual terms:

• \(2a^2b\) with a degree of 3
• \(10a^4b\) with a degree of 5
• \(-9ab\) with a degree of 2
• Constant 6 with a degree of 0

The term with the highest degree among these is \(10a^4b\) with a degree of 5. Thus, the degree of the entire polynomial is 5.
This measure of polynomial degree informs about the curve’s characteristics, such as the maximum number of turns in a graph for scenarios involving graph plotting, and other critical analysis in algebra.