Problem 7 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 4 x^{2}+3 x+2 x+11 x^{2}-3 $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$ 4x^2 + 3x + 2x + 11x^2 - 3 $$ Answer: $$ 15x^2 + 5x - 3 $$

Step 1: Identify like terms

In the given expression, $$ 4x^2 + 3x + 2x + 11x^2 - 3 $$ we can see that we have two terms with \(x^2\): \(4x^2\) and \(11x^2\), and two terms with x: \(3x\) and \(2x\). The constant term is \(-3\).

Step 2: Combine the like terms

As we identified in step 1, we now combine the like terms: - Combining \(4x^2\) and \(11x^2\) gives us \((4+11)x^2 = 15x^2\). - Combining \(3x\) and \(2x\) gives us \((3+2)x = 5x\).

Step 3: Form final simplified expression

By putting the combined terms together, we get the final simplified expression: $$ 15x^2 + 5x - 3 $$

Key Concepts

Combine Like Terms

When simplifying algebraic expressions, an essential step is to combine like terms. Like terms are terms that have the same variables raised to the same power. For example, in the expression \(4x^2 + 3x + 2x + 11x^2 - 3\), the terms \(4x^2\) and \(11x^2\) are like terms because they both contain the variable \(x\) squared.

To combine them, we simply add their coefficients, making \((4+11)x^2 = 15x^2\). Similarly, \(3x\) and \(2x\) are like terms as well, combining to \((3+2)x = 5x\). Combining like terms is a way to consolidate and simplify the expression, making it easier to work with or solve in an equation.

Algebraic Manipulation

Algebraic manipulation encompasses various techniques to transform algebraic expressions into a more useful or simplified form. This includes distributing multiplication over addition, factoring, expanding expressions, and combining like terms, as was done in our exercise.

Key Techniques

For effective algebraic manipulation, it's important to remember the order of operations and properties of real numbers (commutative, associative, distributive). In our example, we practiced manipulation by identifying and combining like terms to significantly simplify the expression. Mastering algebraic manipulation is critical for solving equations, graphing functions, and tackling higher-level math problems.

Elementary Algebra

Elementary algebra is the branch of mathematics that introduces the fundamental concepts of algebra, including variables, expressions, equations, and functions. In elementary algebra, we learn how to work with unknowns represented by letters, which is the foundation for more advanced fields of mathematics. The exercise we solved is a classic example of elementary algebra, where we simplified an algebraic expression by identifying and combining like terms. This practice is critical for developing a deeper understanding of algebraic concepts and preparing for more complex mathematical topics.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial expression to its simplest form. This involves several steps, like combining like terms, factoring, and canceling common factors if present. The process is essential because simplified polynomials are much easier to evaluate, graph, or use in further calculations. Our exercise involved simplifying a polynomial where we combined like terms to reach the final simplified form \(15x^2 + 5x - 3\). Simplifying polynomials is a vital skill in mathematics, often used in calculus, algebra, and other mathematical contexts to understand and solve more complex problems.