Problem 5 - College Algebra with Modeling and Visualization

Question

Combine like terms whenever possible. $$5 x^{2}+8 x+x^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(6x^2 + 8x\).

Step 1: Identify the like terms

In the expression, identify the terms that have the same variable raised to the same power. The given expression is \(5x^2 + 8x + x^2\). Here, \(5x^2\) and \(x^2\) are like terms because they both have the variable \(x\) raised to the power of 2. The term \(8x\) is different because it is \(x\) raised to the power of 1.

Step 2: Combine the like terms

Add the coefficients of the like terms identified in the previous step. For \(5x^2\) and \(x^2\), we have \(5 + 1 = 6\). Thus, combine them to get \(6x^2\). The term \(8x\) remains unchanged as there are no other \(x\) terms to combine with.

Step 3: Write the simplified expression

After combining the like terms, the simplified expression becomes \(6x^2 + 8x\). This is the final result of simplifying the given algebraic expression by combining like terms.

Key Concepts

Combining Like Terms

In algebra, "combining like terms" is an essential concept that helps simplify expressions. This process involves grouping together terms that have identical variable parts.
For instance, in the expression we are discussing, which is \(5x^2 + 8x + x^2\), there are terms like \(5x^2\) and \(x^2\) that can be combined. This is because they both have the variable \(x\) raised to the same power, which is 2.
If you encounter terms like \(6x\) and \(3x\), they are also combinable because they share the same variable, \(x\), even though they don't appear in this particular example.

• Identify terms with the same variable and exponent.
• Add their coefficients together.
• The numerical values (coefficients) are the only parts to change; the variable part remains as it is.

By combining like terms, an algebraic expression becomes simpler and more manageable.

Polynomial Simplification

Polynomial simplification is another important concept in algebra. It refers to reducing a polynomial expression to its simplest form while maintaining its equivalence.
In our example, the polynomial \(5x^2 + 8x + x^2\) requires simplification. Simplification involves not just combining like terms but also organizing the expression clearly.
Here's how polynomial simplification works:

• First, identify and combine like terms, just as we did with \(5x^2\) and \(x^2\), giving us \(6x^2\).
• Keep terms that cannot be combined in their original form, like \(8x\).

The final simplified result of our expression is \(6x^2 + 8x\). This form is not only simplified but also easier to work with in further mathematical operations.

Variables and Coefficients

A solid understanding of variables and coefficients is vital when working with algebraic expressions. These are the building blocks you need to comprehend to effectively manipulate algebraic equations.
Variables are symbols, often letters, that represent numbers or values that can change or vary. For example, in the term \(8x\), \(x\) is a variable.
Coefficients, on the other hand, are the numerical part of the terms which multiply the variables. In the term \(8x\), 8 is the coefficient. If no number is explicitly shown in front of a variable, the coefficient is understood to be 1, like in \(x^2\) where the coefficient is 1.

• Recognize that variables provide a way to express general forms of equations.
• Understand that coefficients indicate how many times to multiply the variable.

Grasping these elements is essential as they form the basis of algebraic operations and help solve equations efficiently.