Problem 29 - Algebra 1: Concepts and Skills

Question

Simplify the expression by combining like terms if possible. If not possible, write already simplified. $$p^{2}+4 p+5 p^{2}-2$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(6 p^{2}+4 p-2\).

Step 1: Identify Like Terms

In the expression \(p^{2}+4 p+5 p^{2}-2\), the like terms are \(p^{2}\) and \(5 p^{2}\) because they both are in the form of \(p^{2}\). Similarly, terms \(4 p\) and \(-2\) are standalone because they do not have similar terms to combine with.

Step 2: Combine Like Terms

To combine the like terms, we simply add or subtract the coefficients as necessary. Adding the like terms \(p^{2}\) and \(5 p^{2}\) gives \(p^{2} + 5 p^{2} = 6 p^{2}\). The terms \(4 p\) and \(-2\) are already standalone, so they remain the same.

Step 3: Write the Simplified Expression

Having combined the like terms, we substitute them back into the expression, resulting in the new, simplified expression \(6 p^{2}+4 p-2\).

Key Concepts

Polynomial Simplification

Simplifying polynomials involves combining like terms to create an expression that is easier to read and work with. At its core, polynomial simplification reduces expressions by aggregating terms that share the same variables and exponents. This process streamlines complex algebraic statements.

In our example, the expression \(p^{2} + 4p + 5p^{2} - 2\) consists of like and unlike terms:

• Like terms: \(p^{2}\) and \(5p^{2}\) because both terms share the same variable \(p\), raised to the power of 2.
• Unlike terms: \(4p\) and \(-2\), which don't have matches with similar variables or exponents.

To simplify, you add the coefficients of like terms. Thus, \(p^{2} + 5p^{2}\) simplifies to \(6p^{2}\). Meanwhile, \(4p\) and \(-2\) remain unchanged since they do not have like terms to combine with. The expression simplifies neatly into \(6p^{2} + 4p - 2\).

Simplifying polynomials is a vital step to make algebraic operations and problem-solving more efficient!

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations such as addition, subtraction, multiplication, and division. Expressions may contain one or more terms. Each term in an algebraic expression is separated by a plus or minus sign.

For instance, the expression \(p^{2} + 4p + 5p^{2} - 2\) has four terms. It combines numerical coefficients and variables with different degrees:

• Term \(p^{2}\)
• Term \(4p\)
• Term \(5p^{2}\)
• Numerical term \(-2\)

The goal is often to simplify these expressions by reducing the number of terms through mathematical operations like combining like terms.

Understanding the nature of algebraic expressions helps in recognizing which terms can be combined to simplify the expression. This concept is fundamental in algebra because it lays the groundwork for more complex operations, functions, and equations.

Coefficients in Algebra

In algebra, coefficients are the numerical or constant multipliers of the terms in an expression. They are the numbers placed directly in front of the variables or features in a term. Understanding coefficients is important when combining like terms, as you only combine terms with the same variable and degree by manipulating their coefficients.

In the expression \(p^{2} + 4p + 5p^{2} - 2\), coefficients can be identified as follows:

• Coefficient of \(p^{2}\) is 1, for the first \(p^{2}\) term.
• Coefficient of \(4p\) is 4.
• Coefficient of \(5p^{2}\) is 5.
• \(-2\) is a constant, sometimes considered a coefficient of \(p^{0}\).

By recognizing these, we can easily combine the like terms. Here, we add the coefficients of the \(p^{2}\) terms: 1 and 5, leading to the coefficient 6 in \(6p^{2}\). This process illustrates the simplification technique fundamental to dealing with algebraic expressions efficiently.