Problem 2 - Elementary and Intermediate Algebra

Question

\(\left(b^{2}-9 b+11\right)-\left(4 b^{2}-14 b\right)\) is the _______ of a trinomial and a binomial.

Step-by-Step Solution

Answer

A trinomial.

Step 1: Identify Terms in the Expression

First, let's identify the terms within each parenthesis. In \(\left(b^{2} - 9b + 11\right)\), we have a trinomial composed of three terms: \(b^{2}\), \(-9b\), and \(11\). In \(\left(4b^{2} - 14b\right)\), we have a binomial with two terms: \(4b^{2}\) and \(-14b\).

Step 2: Distribute the Negative Sign

We'll distribute the negative sign across the second expression. This changes the subtraction into adding the inverse:\[-b^{2} + 9b - 11 - 4b^{2} + 14b. \]

Step 3: Combine Like Terms

Let's combine like terms:- Combine \(-b^{2}\) with \(-4b^{2}\) to get \(-5b^{2}\).- Combine \(9b\) with \(14b\) to get \(23b\).- The constant term is \(-11\).Thus, the resulting expression is \(-5b^{2} + 23b - 11\).

Step 4: Identify the Type of Expression

The expression \(-5b^2 + 23b - 11\) consists of three terms: \(-5b^2\), \(23b\), and \(-11\). Since it has three terms, it is a trinomial.

Key Concepts

Understanding Trinomials

A trinomial is a type of polynomial that consists of exactly three terms. These terms can include variables with coefficients, constants, or a combination of both. Trinomials often appear in algebraic expressions and play a vital role in polynomial operations.

• Each term in a trinomial can have different powers of the same variable or even different variables.
• The structure of a trinomial generally includes terms such as quadratic terms, linear terms, and constant terms.
• For example, in the expression \(b^2 - 9b + 11\), the terms are \(b^2\) (a quadratic term), \(-9b\) (a linear term), and \(+11\) (a constant term).

When performing operations like addition or subtraction with trinomials, ensure that you properly align and combine the terms. The key is to recognize the coefficients and align like terms to simplify the expression. This understanding is crucial for manipulating and simplifying algebraic expressions effectively.

Decoding Binomials

A binomial is a simpler type of polynomial consisting of just two terms. These terms can be quite versatile, involving variables, coefficients, and constants, just like trinomials.

• Binomials typically have a standard format like \(ax^n + bx^m\), where \(a\) and \(b\) are coefficients, and \(n\) and \(m\) are powers of the variable \(x\).
• In the expression \(4b^2 - 14b\), you have the terms \(4b^2\) (a quadratic term) and \(-14b\) (a linear term), making it a binomial.
• Binomials are the building blocks for many algebraic identities and key in polynomial division and multiplication.

When working with binomials, especially in operations like addition or subtraction, it's essential to keep track of the signs and coefficients of each term. This will help you accurately perform operations and achieve the correct simplified results.

Importance of Like Terms

In algebra, recognizing and combining like terms is crucial when simplifying expressions. Like terms have identical variable parts, which means they have the same variable raised to the same power.

• For example, in the given expressions, \(-b^2\) and \(-4b^2\) are like terms because they both involve \(b^2\).
• Similarly, \(9b\) and \(14b\) are like terms since they both contain the variable \(b\).
• Constant terms, such as \(-11\), are also considered like terms because they do not involve any variables.

To simplify an expression like \(-b^2 + 9b - 11 - 4b^2 + 14b\), you need to combine the like terms:\\(-b^2 + 9b - 4b^2 + 14b - 11 = -5b^2 + 23b - 11\).
Combining like terms helps in reducing an expression to its simplest form, making it easier to identify the type of polynomial you are working with and preparing it for further operations.