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Chapter 6: Problem 24

Add or subtract the monomials. $$p q^{2}-4 p-3 q^{2}$$

Short Answer

Expert verified
pq^2 - 4p - 3q^2

Step by step solution

01

Identify like terms

Like terms are terms that have the same variables raised to the same powers. In this problem, we have the terms: \(p q^{2}\), \(-4 p\), and \(-3 q^{2}\). None of these terms are alike because they all have different variables and/or powers.
02

Combine like terms

Since there are no like terms to combine, the expression remains the same: \(p q^{2}-4 p-3 q^{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monomials
A monomial is a type of algebraic expression that consists of a single term. This term can include variables, numbers, and exponents. For example, in the expression **$$p q^{2}-4 p-3 q^{2}$$**, the individual terms **$$p q^{2}$$**, **$$-4 p$$**, and **$$-3 q^{2}$$** are all monomials.
Monomials have the following characteristics:
  • They contain no addition or subtraction symbols within the term.
  • They are made up of numbers (coefficients) and variables.
  • The variables can have exponents.
In our example, **$$p q^{2}$$** has a coefficient of 1, the variable **$$p$$** is raised to the power of 1, and **$$q$$** is squared. Understanding monomials is essential for identifying and working with algebraic expressions.
Like Terms
Like terms are terms within an algebraic expression that have the exact same variables raised to the same powers. Identifying like terms helps simplify expressions by combining them. For example, in the expression **$$p q^{2}-4 p-3 q^{2}$$**, let鈥檚 examine whether any terms are alike:
  • **$$p q^{2}$$**
  • **$$-4 p$$**
  • **$$-3 q^{2}$$**
None of these terms are like terms because they all have different variables and/or exponent combinations. Therefore, they cannot be combined. Always remember to match both the variable and the exponent to determine if terms are alike. Common terms such as **$$2x$$** and **$$3x$$** can be combined because they are both **$$x$$** terms.
Combining Terms
Combining terms is an essential process in simplifying algebraic expressions. It involves adding or subtracting like terms. Here鈥檚 a quick guide:
  • Identify like terms in the expression.
  • Add or subtract the coefficients of these like terms while keeping the variable part unchanged.
  • If there are no like terms, the expression remains unchanged.
In our exercise, since **$$p q^{2}$$**, **$$-4 p$$**, and **$$-3 q^{2}$$** are not like terms, they cannot be combined.
Therefore, the expression stays as **$$p q^{2}-4 p-3 q^{2}$$**.
Combining terms simplifies equations and makes further algebraic manipulation more manageable. Practice identifying and combining like terms to enhance your algebra skills.

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Most popular questions from this chapter

Mixed Practice $$\frac{32 y^{5}}{8 y^{2}}-\frac{60 y^{10}}{5 y^{7}}$$

Divide each polynomial by the binomial. $$\left(x^{2}-3 x-10\right) \div(x+2)$$

Mixed Practice (a) \(q^{5} \cdot q^{3}\) (b) \(\left(q^{5}\right)^{3}\)

Divide each polynomial by the binomial. $$\left(t^{2}-12 t+36\right) \div(t-6)$$

Divide the monomials. $$\frac{\left(10 m^{5} n^{4}\right)\left(5 m^{3} n^{6}\right)}{25 m^{7} n^{5}}$$
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