Problem 21 - Intermediate Algebra

Question

What are the terms of the expression? Give the coefficient of each term. See Objective \(1 .\) $$\frac{11}{12} a^{4}-\frac{3}{4} b^{2}+25 b$$

Step-by-Step Solution

Answer

Terms are \( \frac{11}{12} a^{4}, -\frac{3}{4} b^{2}, \text{and} 25b \) with coefficients \( \frac{11}{12}, -\frac{3}{4}, \text{and} 25 \).

Step 1: Identify the Terms

The expression given is \( \frac{11}{12} a^{4} - \frac{3}{4} b^{2} + 25b \). The terms of an expression are the distinct parts that are connected by addition or subtraction. In this expression, the terms are \( \frac{11}{12} a^{4}, -\frac{3}{4} b^{2}, \text{and} 25b \).

Step 2: Extract Coefficient from Term 1

The first term is \( \frac{11}{12} a^{4} \). The coefficient is the numerical factor in a term. Since \( a^{4} \) is raised to a power without a numerical factor other than the fraction; thus, the coefficient here is \( \frac{11}{12} \).

Step 3: Extract Coefficient from Term 2

The second term is \( -\frac{3}{4} b^{2} \). Here, the coefficient is the numerical part \( -\frac{3}{4} \). This accounts for both the fraction and the negative sign.

Step 4: Extract Coefficient from Term 3

The third term is \( 25b \). The coefficient of \( b \) is \( 25 \), which is the numerical factor associated with \( b \).

Key Concepts

Coefficients

In algebra, coefficients are the numbers in front of the variables in terms. They tell us how many times the term is counted in an expression. For instance, in the expression \( \frac{11}{12} a^{4} - \frac{3}{4} b^{2} + 25b \), we have three terms: \( \frac{11}{12} a^{4} \), \( -\frac{3}{4} b^{2} \), and \( 25b \). These are separated by plus and minus signs.

• The first term \( \frac{11}{12} a^{4} \) has a coefficient \( \frac{11}{12} \), indicating how many units of \( a^{4} \) are in the expression.
• The second term \( -\frac{3}{4} b^{2} \) has a coefficient of \( -\frac{3}{4} \). The negative sign is important as it affects whether the term is added or subtracted.
• Lastly, the term \( 25b \) is straightforward with a coefficient of \( 25 \), reflecting the multiplication of \( 25 \) against \( b \).

Understanding coefficients helps simplify and manipulate algebraic expressions, making it easier to solve equations.

Terms in Algebra

In algebraic expressions, terms are individual parts that make up an expression. These parts are usually connected by addition or subtraction. When we look at \( \frac{11}{12} a^{4} - \frac{3}{4} b^{2} + 25b \), we see three distinct terms. Each term can consist of constants, variables, and exponents.
Terms can be:

• Monomials: A single term like \( 3x \) or \( -\frac{3}{4} b^{2} \).
• Binomials: Two terms that create a single expression, for instance, \( x + 2 \).
• Trinomials: These have three terms, such as \( x^2 - 4x + 4 \).

Identifying terms helps in organizing and simplifying expressions, making complex problems more manageable. Notice how terms are distinct from each other, often characterized by their coefficients, variables, and powers.

Polynomials

In algebra, polynomials are expressions with multiple terms. They are versatile and can consist of constants, variables, exponents, and operations like addition and subtraction. The expression \( \frac{11}{12} a^{4} - \frac{3}{4} b^{2} + 25b \) is a polynomial because:

• It consists of multiple terms.
• Each term has a variable raised to a power, making it a collection of monomials.

Polynomials can also be classified based on the number of terms:

• Monomials: One term, e.g., \( 4x \).
• Binomials: Two terms, e.g., \( x - 7 \).
• Trinomials: Three terms, e.g., \( x^2 + 5x + 6 \).

Polynomials are critical in algebra because they form the basis of many mathematical problems and solutions. They help in expressing equations and can be added, subtracted, or multiplied to find solutions. Understanding polynomials is key to mastering algebra and solving complex equations efficiently.