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Chapter 6: Problem 25

Add or subtract the monomials. $$a^{2} b-4 a-5 a b^{2}$$

Short Answer

Expert verified
\(a^{2} b - 4a - 5a b^{2}\)

Step by step solution

01

Identify the Monomials

The given expression consists of three terms: \(a^{2} b\), \(-4a\), and \(-5a b^{2}\). Each of these terms is a monomial.
02

Check for Like Terms

Like terms in algebra are terms that contain the same variables raised to the same power. Here, none of the terms contain the same combination of variables and exponents: \(a^{2} b\), \(a\), and \(a b^{2}\) are all different.
03

Combine the Terms

Since there are no like terms to combine, the expression remains as is: \(a^{2} b - 4a - 5a b^{2}\).

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Key Concepts

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

like terms
In algebra, 'like terms' are terms that have the exact same variable factors raised to the same power. This makes them compatible for addition or subtraction.
For example:
  • The terms \(3a^{2}b\) and \(7a^{2}b\) are 'like terms' because they both have \(a^{2}b\) as their variables raised to the same powers.
  • However, \(a^{2}b\) and \(ab^{2}\) are not 'like terms' because the exponents of the variables differ.
  • Similarly, \(4x\) and \(4x^{3}\) are also not 'like terms' for the same reason.

Identifying 'like terms' helps us simplify complex expressions.
When adding or subtracting expressions:
  • Combine only 'like terms' by adding or subtracting their coefficients.
  • If there are no 'like terms', the expression remains as it is.
Understanding 'like terms' makes it easier to manage and solve equations.
algebraic expressions
'Algebraic expressions' are mathematical phrases that can contain numbers, variables, and operation symbols (such as +, -, *, /).
They do not include an equal sign, unlike algebraic equations.
For example, \(3a^{2}b + 7a - 5\) is an algebraic expression.
The key components of algebraic expressions are:
  • Terms 鈥� The parts of the expression separated by + or - signs. Each term can be a constant, a variable, or a product of constants and variables. Like in our example: \(a^{2}b, \ -4a, \ and\ -5ab^{2}\)
  • Coefficients 鈥擳he numerical part of the terms. Like \(-4\), \(-5\)
  • Variables 鈥� The letters used to represent numbers. For instance, a, b in our case.
The main goal when working with algebraic expressions is to simplify or rearrange them to facilitate easier calculations or problem solving.
monomials
A 'monomial' is a type of algebraic expression that has only one term. It responds to the product, typically, of constants and variables.
For example, \(5x^{3}y\), \(7a\), and \(12b^{2}c\) are all monomials.
The essential characteristics of monomials include:
  • Single term 鈥� Unlike an algebraic expression, a monomial does not have addition or subtraction signs connecting multiple terms.
  • Product of constants and variables 鈥� Monomials are formed by multiplying constants (numbers) and variables (letters).
Monomials play a crucial role in polynomial expressions and are fundamental components in algebra. When working with monomials:
  • Identify and combine 'like terms' in more complex expressions.
  • Recognize that each monomial term can only be added or subtracted with another identical variable and exponent term.
Understanding what monomials are helps simplify many algebra problems.

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Most popular questions from this chapter

Multiply. $$(w-6) \cdot 8$$

Square each binomial using the Binomial Squares Pattern. $$(2 y-3 z)^{2}$$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$\left(6 m^{3}-4 n^{5}\right)\left(6 m^{3}+4 n^{5}\right)$$

Add or subtract the polynomials. $$\left(p^{2}-6 p-18\right)+\left(2 p^{2}+11\right)$$

Convert each number to decimal form. $$1.6 \times 10^{10}$$
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