Problem 55 - Elementary and Intermediate Algebra

Question

Subtract the polynomials. $$ \left(5 a b+2 b^{2}\right)-\left(2+a b+b^{2}\right) $$

Step-by-Step Solution

Answer

\(4ab + b^2 - 2\)

Step 1: Distribute the Negative Sign

First, distribute the negative sign across the second polynomial \((2 + ab + b^2)\). This means you change the sign of each term in the second polynomial: \(-2 - ab - b^2\).

Step 2: Combine Like Terms

Now combine like terms from both polynomials. Start with the terms involving \(ab\): \(5ab - ab = 4ab\). For the \(b^2\) terms: \(2b^2 - b^2 = b^2\). Finally, handle the constant terms: \(0 - 2 = -2\).

Step 3: Write the Result

After combining, you have the expression: \(4ab + b^2 - 2\). This is the result after subtracting the polynomials.

Key Concepts

Distribute Negative Sign

When subtracting polynomial expressions, an essential first step is to distribute the negative sign properly. This process involves applying the negative sign to each term within the polynomial you’re subtracting. For instance, in the expression \[(5ab + 2b^2) - (2 + ab + b^2),\]focus on converting the second polynomial correctly.
By distributing the negative sign across \[(2 + ab + b^2),\]we alter every term: \[2 \to -2, \ ab \to -ab, \ b^2 \to -b^2.\]This step is critical because neglecting to change the signs can lead to errors in further calculations. It helps to think of subtracting a polynomial as adding the opposite. Always double-check your work by ensuring every term in the second polynomial is modified by the negative sign.

Combine Like Terms

Once we have distributed the negative sign successfully, the next action is to combine like terms.
Like terms in polynomial expressions are those that have the same variable parts. This means they share the same variables raised to the same powers. In our subtraction exercise:

• The terms involving \(ab\) : \(5ab - ab = 4ab\).
• For terms involving \(b^2\) : \(2b^2 - b^2 = b^2\).
• And the constants: \(0 - 2 = -2\).

This process ensures that each set of like terms is simplified into a single term, making the polynomial expression easier to interpret or use in further math operations. Combining terms correctly is crucial because it reflects the accurate simplification of polynomial operations.

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables, coefficients, and exponents, organized as the sum of terms. Each term in a polynomial is a product of a number (coefficient) and a variable raised to a power. Understanding how to manipulate these expressions is foundational in algebra.
For example, consider the expression \(5ab + 2b^2\).
The term \(5ab\) is a product of the coefficient 5 and the variables \(a\) and \(b\). The term \(2b^2\) involves the coefficient 2 and the variable \(b\) squared.
Recognizing the structure of these expressions allows us to perform operations like addition, subtraction, and expansion accurately. Mastering operations on polynomial expressions is not only crucial for solving equations but also forms a building block for further study in calculus and other higher-level mathematical fields. Understanding polynomials paves the way for a deeper insight into how variables interact within mathematics.