Problem 6 Combine like terms and simplify.... [FREE SOLUTION]

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Chapter 2: Problem 6

Combine like terms and simplify. $$11-k^{2}+12 k^{2}-3+6 k^{2}$$

Short Answer

Expert verified

The simplified expression after combining like terms is: $8 + 17k^2$.

Step by step solution

(Step 1: Combine constant terms)

(Add 11 and -3 to get the combined constant term. 11 + (-3) = 8 )

(Step 2: Combine $k^2$ terms)

(Add the coefficients of the $k^2$ terms: 12+ (-1) + 6 = 17. So the combined $k^2$ term is 17$k^2$.)

(Step 3: Write the simplified expression)

(Combine the results from steps 1 and 2. The simplified expression is 8 + 17$k^2$.) So the simplified expression after combining like terms is: \[8 + 17k^2\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms

When working with algebraic expressions, one of the most crucial processes is combining like terms. Like terms are terms that have exactly the same variable factors. For instance, in the expression $11-k^{2}+12k^{2}-3+6k^{2}$, the terms involving $k^2$ (i.e., $-k^{2}$, $12k^{2}$, and $6k^{2}$) are considered like terms.
To combine like terms:

Add or subtract the coefficients of the terms. In our example, we have three $k^2$ terms, so we combine them by focusing on their coefficients: $12 - 1 + 6$, which equals $17$.
For constant numbers, which are also like terms but without variables, simply operate with them directly. Here, combine 11 and -3 to yield 8.

Combining like terms simplifies the expression by grouping the terms with similar characteristics, making it easier to solve or further manipulate.

Polynomials

Polynomials are expressions composed of variables, coefficients, and constants, combined using addition, subtraction, multiplication, and non-negative integer exponents. In the expression $8 + 17k^2$, it is a polynomial consisting of two terms: a constant term (8) and a term involving $k^2$.
Some key points about polynomials include:

A polynomial can have multiple terms, but each term can only have whole number exponents.
The degree of a polynomial is determined by the highest power of the variable present in the expression. In our example, the highest power is 2, so its degree is 2.
Polynomials can be classified based on the number of terms. A single term is called a monomial, two terms a binomial, and three terms a trinomial.

Understanding polynomials is fundamental in algebra as it allows students to identify and handle expressions systematically.

Simplification

Simplification in algebra involves transforming an expression into its simplest form through various operations, such as combining like terms or reducing fractions.
The goal of simplification is to make the expression easier to work with or understand without changing its value.

Start by combining like terms as we have done in previous sections. This step often reduces the complexity of an expression.
After combining, reorder terms (if necessary) to follow a conventional format, often arranging terms from highest to lowest degree, although in this example it isn't needed since $17k^2$ is clearly determined as the higher degree term.
Check the expression for any common factors or further reductions, though this example doesn't require those steps.

To master algebra, it is essential to practice simplification routinely, as it helps build the foundation for solving equations and understanding more complex algebraic structures.

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91影视

Combine like terms and simplify. $$11-k^{2}+12 k^{2}-3+6 k^{2}$$

Short Answer

Step by step solution

(Step 1: Combine constant terms)

(Step 2: Combine \(k^2\) terms)

(Step 3: Write the simplified expression)

Key Concepts

Combining Like Terms

Polynomials

Simplification

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