Problem 57 - Algebra A Combined Function

Question

Simplify each polynomial by combining any like terms. See Examples 13 and 14. $$ 14 y^{3}-9+3 a^{2} b^{2}-10-19 b^{2} a^{2} $$

Step-by-Step Solution

Answer

The simplified polynomial is \(14y^3 - 16a^2b^2 - 19\).

Step 1: Identify Like Terms

Look at the polynomial: \(14y^3 - 9 + 3a^2b^2 - 10 - 19b^2a^2\). We need to identify like terms, which are terms with the same variables raised to the same powers.

Step 2: Rearrange the Terms

Rearrange the terms to group like terms together: \(14y^3 + 3a^2b^2 - 19b^2a^2 - 9 - 10\). This helps us easily combine them in the next step.

Step 3: Combine Like Terms for \(a^2b^2\)

Combine the terms with \(a^2b^2\): \(3a^2b^2 - 19a^2b^2 = (3 - 19)a^2b^2 = -16a^2b^2\).

Step 4: Combine the Constant Terms

Combine the constant terms: \(-9 - 10 = -19\).

Step 5: Write the Simplified Polynomial

After combining like terms, the simplified polynomial is: \(14y^3 - 16a^2b^2 - 19\).

Key Concepts

Like Terms

When simplifying polynomials, one of the most important steps is identifying "like terms." Like terms are terms that have the exact same variable parts, including the same exponents. For instance, in the expression given, the terms \(3a^2b^2\) and \(-19b^2a^2\) are considered like terms because they both contain the variables \(a^2\) and \(b^2\). The order of the variables does not matter; only their presence and their exponents do.

When you have like terms, you can combine them by adding or subtracting their coefficients. This is a key process in simplifying polynomials. If terms are not identical in their variable parts, they cannot be combined. Recognizing and combining like terms helps in reducing a polynomial to its simplest form.

Polynomials

Polynomials are algebraic expressions that contain one or more terms. Each term in a polynomial is made up of a constant, a variable, or a combination of both, and these terms are connected by addition or subtraction.

A polynomial can have any number of terms, but when simplifying, the goal is often to reduce it to the minimal number of terms possible by combining like terms. Polynomials can take many forms, such as quadratic, cubic, or higher degrees, based on the highest power of the variable present.

In our exercise, we started with a polynomial that had five terms and after simplification, reduced it to three terms. This usually makes the polynomial easier to interpret and work with in further algebraic operations.

Combining Terms

Combining terms in a polynomial involves the process of adding or subtracting coefficients of like terms to simplify the expression. You assess each term to check if there are others with similar variable components. If there are, you add or subtract their coefficients.

For example, in the given polynomial, the terms with \(a^2b^2\) need to be combined. Their coefficients are 3 and -19, respectively. The combined term results in \(-16a^2b^2\) after adding the coefficients together (\(3 + (-19)\)). This simplification step is crucial for reducing the polynomial to its simplest form, making it less complex, and easier to use in further computations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Polynomials are a specific type of algebraic expression used frequently in algebra and calculus. Each algebraic expression comprises one or more terms that are included in calculations by operations such as addition, subtraction, multiplication, and division.

Working with algebraic expressions involves understanding each term's components and how they interact within the overall structure. The simplification or manipulation of these expressions is a fundamental skill in algebra, allowing for more manageable equations and solutions.

In our exercise, you worked with an algebraic expression in the form of a polynomial. Understanding how to identify like terms and combine them effectively simplifies the expression, enhancing one's ability to work with algebraic equations effortlessly.