Problem 33 - Algebra 2

Question

Simplify. Classify each result by number of terms. $$ \left(2 c^{2}+9\right)-\left(3 c^{2}-7\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( -c^{2}+16 \) and it is a binomial.

Step 1 - Expand the expression

Remove the parentheses and take into account the minus sign before the second parentheses which will change the sign of each term inside it.

Step 2 - Combine like terms

Combine like terms by adding or subtracting the coefficients of the same power of c.

Step 3 - Classify the result by number of terms

Once like terms are combined, count the number of terms to classify the result.

Key Concepts

Polynomial Arithmetic

Understanding polynomial arithmetic is crucial when you're simplifying algebraic expressions. Polynomial arithmetic involves adding, subtracting, multiplying, and sometimes dividing polynomials. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, in the expression \(2 c^{2}+9\), \(2 c^{2}\) and 9 are terms of a polynomial where \(c\) is the variable, 2 is the coefficient of the first term, and 9 is a constant term.

During the arithmetic process, especially in addition or subtraction, it's important to recognize that only like terms can be combined. Also, when you introduce subtraction, as in the original exercise \(\left(2 c^{2}+9\right)-\left(3 c^{2}-7\right)\), you must distribute the minus sign across the terms in the second polynomial, effectively changing their signs before combining like terms. This step is foundational and leads to correct simplification.

Combining Like Terms

The process of combining like terms is, in essence, simplifying an algebraic expression so that it has the fewest terms possible. Like terms are terms that have the exact same variables raised to the same power. Only the coefficients of these like terms can differ. For maintaining the integrity of the expression, it's crucial to add or subtract only the coefficients for like terms.

In our exercise, \(2 c^{2}\) and \(3 c^{2}\) are like terms. To combine them, you would handle only the coefficients: \(2 - 3 = -1\), resulting in \( -1c^{2}\) when they are combined after considering the subtraction indicated in the original problem. This principle reduces the complexity of lengthy polynomials and is the cornerstone of algebraic manipulation.

Why Combine Like Terms?

• It simplifies expressions to their most basic form.
• It sets the stage for further operations, such as solving equations.
• It helps in identifying the degree and structure of the polynomial.

Classifying Polynomial Terms

Once you've simplified an algebraic expression by combining like terms, it's insightful to classify the polynomial based on the number of terms. Polynomials can be monomials (single term), binomials (two terms), trinomials (three terms), or have many terms (multinomials).

In our example, after simplifying, we end up with \( -1c^{2} + 16\), which is a binomial since it is composed of two terms. Classification not only assists in understanding the nature of the polynomial but also is useful in determining which methods of factoring or solving might be most applicable. For instance, binomials may be factored using difference of squares or other applicable techniques depending on the nature of the terms involved.

Important Aspects When Classifying:

• Count the number of terms after simplification.
• Take note of variables and their powers - they don't affect classification by number of terms.
• Use classification to predict and apply further algebraic methods.