Problem 17 - Algebra for College Students

Question

Add the given polynomials. \(12 a^{2} b^{2}-9 a b\) and \(5 a^{2} b^{2}+4 a b\)

Step-by-Step Solution

Answer

The sum is \(17a^2b^2 - 5ab\).

Step 1: Understand the Problem

We have two polynomials that we need to add: \(12a^2b^2 - 9ab\) and \(5a^2b^2 + 4ab\). To find the sum, we must combine like terms, which are terms with the same variables raised to the same powers.

Step 2: Identify Like Terms

Identify the like terms in both polynomials. \(12a^2b^2\) and \(5a^2b^2\) are like terms because they both have \(a^2b^2\). Similarly, \(-9ab\) and \(4ab\) are like terms because they both have \(ab\).

Step 3: Add Like Terms

Add the coefficients of the like terms: For \(a^2b^2\) terms: \(12 + 5 = 17\). For \(ab\) terms: \(-9 + 4 = -5\).So, the sum of the polynomials is \(17a^2b^2 - 5ab\).

Step 4: Write the Result

The result of adding the polynomials is \(17a^2b^2 - 5ab\).

Key Concepts

Like Terms

When dealing with polynomial addition, recognizing like terms is essential. Like terms refer to terms within an expression that share the same variable part. This means the variables and their exponents are identical in each term.
For instance, in the polynomials given in our original exercise, the terms \(12a^2b^2\) and \(5a^2b^2\) are considered like terms because they both contain the identical variable combination \(a^2b^2\).
Identifying like terms allows us to effectively combine these terms by focusing only on their coefficients. It's a crucial step in simplifying expressions correctly.

Coefficients

In algebra, coefficients are the numerical factors of terms. They are the numbers located in front of the variables in an expression. Coefficients define how many times a term is multiplied by the given variable pattern.
For example, in the term \(12a^2b^2\), the number 12 is the coefficient. It tells us how many 'groups' of \(a^2b^2\) we have. Similarly, in the term \(-9ab\), the coefficient is -9.
When adding polynomials, it's the coefficients of like terms that we add together. This simplifies the polynomial while keeping the variable part the same.

Combining Polynomials

Combining polynomials involves adding or subtracting them by using their like terms. This process consists of:

• Listing each term from the polynomials involved.
• Identifying the like terms by matching the variables and their exponents.
• Adding or subtracting the coefficients of these like terms.

In our example, \(12a^2b^2 + 5a^2b^2\) combines to form \(17a^2b^2\), and \(-9ab + 4ab\) simplifies to \(-5ab\).
This consolidation of terms makes the expression easier to use in subsequent calculations or in solving equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They represent general values and relationships in mathematics.
Expressions do not include an equal sign, unlike equations. For example, the polynomials \(12a^2b^2 - 9ab\) and \(5a^2b^2 + 4ab\) are algebraic expressions. They form part of the basis for building more complex mathematical models.
Understanding these expressions involves recognizing different components like terms, coefficients, and variables. Mastery of algebraic expressions paves the way for solving equations and understanding functions in more advanced mathematics.