Problem 32 - Elementary Algebra

Question

\(3 p^{2}+p-10+2 p^{2}-2 p+13\)

Step-by-Step Solution

Answer

The simplified expression is 5p^2 - p + 3.

Step 1: Combine Like Terms

Combine the terms that have the same variable. Here, combine the terms with the variable raised to the power of 2, the terms with the variable raised to the power of 1, and finally combine the constant terms.

Step 2: Combine Quadratic Terms

Add the coefficients of the quadratic terms: 3 and 2. Hence, 3p^2 + 2p^2 = 5p^2

Step 3: Combine Linear Terms

Add the coefficients of the linear terms: 1 and -2. Hence, p - 2p = -p

Step 4: Combine Constants

Add the constant terms: -10 and 13. Hence, -10 + 13 = 3

Step 5: Form the Final Expression

Combine all simplified terms to form the final expression: 5pd^2 - p + 3

Key Concepts

quadratic terms

Quadratic terms are the parts of an algebraic expression where the variable is raised to the power of 2. In our example, these terms are represented as \(3p^2\) and \(2p^2\). When combining quadratic terms, you add their coefficients together. Here, the coefficients are 3 and 2. So, the calculation would be: 3 + 2 = 5. Therefore, when combined, the quadratic terms form \(5p^2\).

linear terms

Linear terms are the parts of an algebraic expression where the variable is raised to the power of 1. For our given exercise, these terms are \(p\) and \(-2p\). When you combine linear terms, you also add their coefficients. For linear terms in our problem, the coefficients are 1 and -2. So, the calculation is: 1 - 2 = -1. Therefore, the combined linear terms result in \(-p\).

constants

Constants are the number terms in an algebraic expression that do not contain variables. In the provided exercise, the constants are -10 and 13. To combine constants, you simply add them. So, -10 + 13 equals 3. Therefore, our expression, once constants are combined, includes the term 3.