Problem 72 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 5 x^{2}+2 x-3-7 x^{2}-3 x-4-2 x^{2}-11 $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: $$ 5x^{2}+2x-3-7x^{2}-3x-4-2x^{2}-11 $$ Answer: $$ -4x^{2} -1x -18

Step 1: Identify like terms

Look for the terms with the same variable raised to the same power and constants. In this expression, we have: $$ 5 x^{2}+2 x-3-7 x^{2}-3 x-4-2 x^{2}-11 $$ Like terms are: 1. \(5x^{2}, -7x^{2}, -2x^{2}\) 2. \(2x, -3x\) 3. \(-3, -4, -11\)

Step 2: Combine like terms

Add or subtract the like terms: 1. Combine \(5x^{2}, -7x^{2}, -2x^{2}\): $$ 5x^{2} - 7x^{2} -2x^{2} = (5 - 7 - 2)x^{2} = -4x^{2} $$ 2. Combine \(2x, -3x\): $$ 2x - 3x = (2 - 3)x = -1x $$ 3. Combine constants \(-3, -4, -11\): $$ -3 - 4 - 11= -18 $$

Step 3: Combine all the simplified terms

Add together the simplified terms (-4x², -1x, and -18) to form the final simplified algebraic expression: $$ -4x^{2} -1x -18 $$ So, the simplified algebraic expression for the given problem is: $$ -4x^{2} -1x -18 $$

Key Concepts

Combining Like Terms

Understanding how to combine like terms is essential when simplifying algebraic expressions. It involves adding or subtracting terms that have exactly the same variable part. For instance, consider the terms involving the variable 'x' with the same exponent. In the exercise ewline ewline \[5 x^{2} + 2 x - 3 - 7 x^{2} - 3 x - 4 - 2 x^{2} - 11\] ewline ewline the terms \(5x^{2}, -7x^{2}, -2x^{2}\) all have the variable 'x' raised to the power of 2, hence they can be combined by adding their coefficients: ewline ewline \[ (5 - 7 - 2)x^{2} = -4x^{2} \] ewline ewline Similarly, to combine the terms \(2x, -3x\), since they are both 'x' to the power of 1, we combine the coefficients: ewline ewline \[ (2 - 3)x = -1x \] ewline ewline It's like grouping fruits; you wouldn't mix apples and oranges when counting how many of each you have. Remember, terms that do not share the same variable and exponent are not like terms and cannot be combined in this way.

Algebraic Expression Simplification

The goal of algebraic expression simplification is to rewrite expressions in the simplest form possible, making them easier to interpret and work with. Simplification can involve several steps, including combining like terms, as we have discussed, as well as applying the distributive property and canceling terms where applicable. ewline ewline Once you have combined like terms by adding or subtracting them, other aspects of simplification may come into play. For constant terms in the exercise provided: ewline ewline \[ -3 - 4 - 11\] ewline ewline these can be added together because they are like terms (they don’t have any variables). When you simplify, the expression compresses to ewline ewline \[ -4x^{2} - 1x - 18 \] ewline ewline where each type of term is represented only once. Simplification is not only about making expressions shorter; it ensures they are presented in a form that is widely understood and easily used for further algebraic manipulation, such as solving equations or factoring.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the general properties of numbers and the relationships between them using symbols and letters to represent numbers, variables, and arithmetic operations. In the context of our exercise, elementary algebra comes into play when identifying and combining like terms, applying the rules of addition and subtraction to both constant and variable terms, and understanding the fundamental operations on polynomials. ewline ewline One of the foundational skills in elementary algebra is recognizing patterns and structure in expressions, such as the similarity in terms of variables and exponents. When simplifying the expression ewline ewline \[ 5 x^{2}+2 x-3-7 x^{2}-3 x-4-2 x^{2}-11 \] ewline ewline we used those basic principles to combine and simplify. Mastery of elementary algebra is crucial for progression to more advanced math such as algebra II, calculus, and beyond. With practice, the simplification of algebraic expressions becomes a straightforward and invaluable process.