Problem 47 - Algebra A Combined Function

Question

Identify the degrees of the terms and the degree of the polynomial. See Example 12. $$ 9 a b-6 a+5 b-3 $$

Step-by-Step Solution

Answer

The degrees of terms are 2, 1, 1, 0; polynomial degree is 2.

Step 1: Identify the Terms

The given expression is \(9ab - 6a + 5b - 3\). We can identify four terms in this polynomial:1. \(9ab\)2. \(-6a\)3. \(5b\)4. \(-3\)

Step 2: Determine the Degree of Each Term

The degree of a term is the sum of the exponents of the variables within that term.- For \(9ab\): the degree is \(1 + 1 = 2\) because \(a\) and \(b\) are both raised to the power of \(1\).- For \(-6a\): the degree is \(1\) because \(a\) is raised to the power of \(1\).- For \(5b\): the degree is \(1\) because \(b\) is raised to the power of \(1\).- For \(-3\): the degree is \(0\) because it is a constant term with no variable.

Step 3: Identify the Degree of the Polynomial

The degree of a polynomial is the highest degree of its terms. Among the degrees calculated, the terms have the degrees: \(2, 1, 1, 0\). The highest of these is \(2\). Thus, the degree of the polynomial \(9ab - 6a + 5b - 3\) is \(2\).

Key Concepts

Degree of a Term

The degree of a term is an important concept when dealing with polynomials. It refers to the highest power of the variable(s) in that term. In simple terms, it tells us the sum of the exponents of all the variables within the term.

• For the term \(9ab\), each variable \(a\) and \(b\) is raised to the power of \(1\), so the degree is \(1 + 1 = 2\).
• For the term \(-6a\), the variable \(a\) is raised to the power of \(1\), giving it a degree of \(1\).
• In the term \(5b\), the variable \(b\) is at power \(1\), making its degree \(1\).
• Finally, the term \(-3\) has no variables, so its degree is \(0\).

Understanding the degree of each term helps in identifying the degree of the polynomial later on.

Polynomial Terms

A polynomial is made up of 'terms', and these are essentially the building blocks of the polynomial expression. Each term contains a coefficient (a numerical factor), variables, and exponents:

• The term \(9ab\) is composed of the number \(9\) (coefficient) and the variables \(a\) and \(b\).
• For the term \(-6a\), the coefficient is \(-6\) and the variable is \(a\).
• The term \(5b\) includes the coefficient \(5\) and the variable \(b\).
• \(-3\) is a constant term, meaning it has no variables.

These terms are connected by addition or subtraction to form the polynomial. Recognizing and understanding each component allows us to break down and analyze the polynomial efficiently.

Exponents in Polynomials

Exponents are crucial in understanding polynomials as they indicate the power to which a variable is raised. In a term, exponents help determine the term's degree:

• The term \(9ab\) involves the variables \(a\) and \(b\), each to the exponent of \(1\). Therefore, exponents are the basis for defining this term's degree \( (1 + 1 = 2) \).
• In the case of \(-6a\), the exponent for \(a\) is \(1\).
• For \(5b\), the exponent of \(b\) is also \(1\).
• \(-3\) lacks any variable part, so exponents do not apply here, making its degree \(0\).

Knowing the exponents of each term is pivotal when aiming to determine the overall degree of the polynomial.

Constant Term in Polynomials

The constant term in a polynomial is a term that has no variable associated with it. It remains "constant" throughout any calculations since it does not change with variable values.

In the expression \(9ab - 6a + 5b - 3\), the term \(-3\) is the constant term. Since this term lacks a variable, it has a degree of \(0\). Constants are a significant aspect of polynomials, as they are present in many polynomial calculations but do not impact the degree of the polynomial.

Understanding constant terms helps clarify their role both in calculations and in determining the complexity or 'degree' of algebraic expressions.