How To Find The Sum Of Monomials A Step By Step Guide

In mathematics, a monomial is an expression that consists of a single term. This term can be a number, a variable, or a product of numbers and variables. Understanding how to add monomials is a fundamental skill in algebra. This article provides a comprehensive guide on how to find the sum of various monomials, complete with detailed explanations and examples. Whether you're a student looking to improve your algebra skills or just someone interested in mathematics, this guide will help you grasp the concepts and techniques needed to solve these types of problems.

1. 4z + 21z

To find the sum of monomials 4z and 21z, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, both terms have the variable 'z' raised to the power of 1. Therefore, we can add their coefficients directly.

Combining like terms involves adding the coefficients of the terms while keeping the variable part the same. The coefficients are the numerical parts of the terms.

For the expression 4z + 21z, the coefficients are 4 and 21. Adding these coefficients gives us:

4 + 21 = 25

So, the sum of 4z and 21z is 25z.

This process is similar to combining any like quantities. For example, if you have 4 apples and you add 21 apples, you will have 25 apples in total. In the same way, 4z plus 21z equals 25z. This basic principle is crucial in algebra for simplifying expressions and solving equations.

When dealing with more complex expressions, always ensure that you are combining only like terms. For instance, you cannot add terms like z and z² because they have different powers of the variable z. Understanding this distinction is vital for accurate algebraic manipulations.

In summary, to add the monomials 4z and 21z, simply add their coefficients: 4 + 21 = 25. Therefore, the final answer is 25z. This example illustrates a fundamental concept in algebra: the addition of like terms.

2. a - b

In this case, we are asked to find the sum of the monomials 'a' and '-b'. It's important to recognize that 'a' and 'b' are different variables, and therefore, they are not like terms. Like terms must have the same variable raised to the same power. Since 'a' and 'b' are distinct variables, we cannot combine them in the same way we combined 'z' terms in the previous example.

When variables are different, the expression remains as is. There is no simplification possible beyond stating the expression. This concept is crucial in algebra because it highlights the importance of identifying like terms before attempting to simplify or combine them.

The expression 'a - b' represents the difference between the values of 'a' and 'b'. Without knowing the specific values of 'a' and 'b', we cannot simplify this expression any further. This is a fundamental principle in algebra: unlike terms cannot be combined.

Imagine 'a' representing the number of apples you have and 'b' representing the number of bananas. You cannot simply add apples and bananas together and get a combined quantity of a single fruit. They remain distinct quantities.

Similarly, in algebra, 'a' and '-b' remain distinct terms. The expression 'a - b' is already in its simplest form. There are no like terms to combine, and thus, no further simplification is possible. This underscores the significance of understanding what constitutes a like term in algebraic expressions.

In conclusion, the sum of the monomials 'a' and '-b' is simply 'a - b'. This expression cannot be simplified further because 'a' and 'b' are different variables and therefore, unlike terms.

3. (-5h) + (-3h) + (-10h)

To find the sum of the monomials (-5h), (-3h), and (-10h), we need to combine these like terms. In this case, all three terms contain the variable 'h' raised to the power of 1, making them like terms. Combining like terms involves adding their coefficients while keeping the variable part the same.

The coefficients in this expression are -5, -3, and -10. We need to add these coefficients together:

-5 + (-3) + (-10)

Adding -5 and -3 gives us:

-5 + (-3) = -8

Now, we add -8 to -10:

-8 + (-10) = -18

So, the sum of the coefficients is -18. Since all the terms have the variable 'h', we multiply the sum of the coefficients by 'h' to get the final result:

-18h

Therefore, the sum of the monomials (-5h), (-3h), and (-10h) is -18h. This process demonstrates how to combine multiple like terms in an algebraic expression. The key is to identify the like terms, add their coefficients, and then multiply the result by the common variable.

This concept is similar to adding multiple quantities of the same item. For instance, if you have -5 units of 'h', -3 units of 'h', and -10 units of 'h', you would have a total of -18 units of 'h'. The negative sign indicates a deficit or a subtraction from a whole.

In summary, adding the monomials (-5h), (-3h), and (-10h) involves adding their coefficients: -5 + (-3) + (-10) = -18. The final sum is -18h. This example highlights the importance of correctly handling negative numbers when combining like terms in algebra.

4. 12b² + (-12b²)

To find the sum of the monomials 12b² and (-12b²), we need to combine these like terms. Like terms are terms that have the same variable raised to the same power. In this case, both terms have the variable 'b' raised to the power of 2, which means they are like terms.

When combining like terms, we add their coefficients. The coefficient of the first term, 12b², is 12, and the coefficient of the second term, (-12b²), is -12. So, we need to add 12 and -12:

12 + (-12)

Adding a number and its negative always results in zero:

12 + (-12) = 0

Since the sum of the coefficients is 0, the sum of the monomials is 0 multiplied by b², which is simply 0.

0 * b² = 0

Therefore, the sum of 12b² and (-12b²) is 0. This example illustrates an important concept in algebra: when you add a term and its additive inverse (the negative of the term), the result is always zero. This is a fundamental property of numbers and algebraic expressions.

Consider this in a real-world context: if you have 12 units of something and you subtract 12 units of the same thing, you are left with nothing. Similarly, in algebra, adding 12b² and -12b² results in a cancellation, leaving zero.

In summary, to find the sum of 12b² and (-12b²), add their coefficients: 12 + (-12) = 0. The final result is 0. This example showcases the concept of additive inverses in algebraic expressions.

5. 2x + 4y + 2z

In this problem, we are asked to find the sum of the monomials 2x, 4y, and 2z. These terms involve different variables: x, y, and z. Since the variables are different, these terms are not like terms. Like terms must have the same variable raised to the same power.

When terms are not like terms, they cannot be combined or simplified further. The expression remains as is, with each term separate. This concept is crucial in algebra because it dictates when terms can be combined and when they must remain distinct.

Imagine you have 2 apples, 4 bananas, and 2 oranges. You cannot add these fruits together and get a single quantity of a single type of fruit. They remain separate and distinct quantities.

Similarly, in algebra, the terms 2x, 4y, and 2z cannot be combined. The expression 2x + 4y + 2z is already in its simplest form. There are no like terms to combine, and thus, no further simplification is possible. This underscores the significance of understanding what constitutes a like term in algebraic expressions.

The expression 2x + 4y + 2z represents the sum of three different quantities, each associated with a different variable. Without knowing the specific values of x, y, and z, we cannot simplify this expression any further.

In conclusion, the sum of the monomials 2x, 4y, and 2z is simply 2x + 4y + 2z. This expression cannot be simplified further because x, y, and z are different variables and therefore, unlike terms.

6. 6s - 2s - 8s

To find the sum of the monomials 6s, -2s, and -8s, we need to combine these like terms. In this case, all three terms contain the variable 's' raised to the power of 1, which means they are like terms. Combining like terms involves adding their coefficients while keeping the variable part the same.

The coefficients in this expression are 6, -2, and -8. We need to add these coefficients together:

6 + (-2) + (-8)

First, let's add 6 and -2:

6 + (-2) = 4

Now, we add 4 to -8:

4 + (-8) = -4

So, the sum of the coefficients is -4. Since all the terms have the variable 's', we multiply the sum of the coefficients by 's' to get the final result:

-4s

Therefore, the sum of the monomials 6s, -2s, and -8s is -4s. This process demonstrates how to combine multiple like terms in an algebraic expression. The key is to identify the like terms, add their coefficients, and then multiply the result by the common variable.

This concept is similar to adding and subtracting quantities of the same item. For instance, if you have 6 units of 's', you subtract 2 units of 's', and then you subtract 8 units of 's', you would have a total of -4 units of 's'. The negative sign indicates a deficit or a subtraction from a whole.

In summary, adding the monomials 6s, -2s, and -8s involves adding their coefficients: 6 + (-2) + (-8) = -4. The final sum is -4s. This example highlights the importance of correctly handling negative numbers when combining like terms in algebra.

7. 12b² + 4b + 3b² + 7b

To find the sum of the monomials 12b², 4b, 3b², and 7b, we need to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two types of like terms: terms with b² and terms with b.

First, let's identify the terms with b²: 12b² and 3b². These are like terms because they both have the variable 'b' raised to the power of 2. We can add their coefficients:

12 + 3 = 15

So, the sum of these terms is 15b².

Next, let's identify the terms with b: 4b and 7b. These are like terms because they both have the variable 'b' raised to the power of 1. We can add their coefficients:

4 + 7 = 11

So, the sum of these terms is 11b.

Now, we combine the sums of the like terms:

15b² + 11b

These terms cannot be combined further because b² and b are not like terms (they have different powers of 'b'). Therefore, the simplified expression is:

15b² + 11b

This example demonstrates the process of combining like terms in a more complex algebraic expression. The key is to first identify the like terms, add their coefficients, and then write the resulting expression. Terms with different variables or different powers of the same variable cannot be combined.

Consider this in a real-world context: if you have 12 square units of something and you add 3 square units, you have a total of 15 square units. Similarly, if you have 4 units of something and you add 7 units, you have 11 units. These are distinct quantities that cannot be combined into a single unit.

In summary, to find the sum of 12b² + 4b + 3b² + 7b, we combine like terms: 12b² + 3b² = 15b² and 4b + 7b = 11b. The final expression is 15b² + 11b.

8. (2a + 6) - (5b + 2)

To find the sum of the monomials represented by the expression (2a + 6) - (5b + 2), we need to simplify the expression by distributing the negative sign and combining like terms. This involves understanding how to handle parentheses and negative signs in algebraic expressions.

First, we distribute the negative sign to the terms inside the second set of parentheses:

(2a + 6) - (5b + 2) = 2a + 6 - 5b - 2

Now, we look for like terms. In this expression, the like terms are the constants 6 and -2. The terms 2a and -5b are not like terms because they involve different variables (a and b).

Combine the like terms (the constants):

6 - 2 = 4

Now, we rewrite the expression with the combined like terms:

2a - 5b + 4

The terms 2a, -5b, and 4 are not like terms, so we cannot combine them further. Therefore, the simplified expression is:

2a - 5b + 4

This example demonstrates the importance of distributing negative signs and identifying like terms when simplifying algebraic expressions. The process involves removing parentheses, combining constants, and ensuring that only like terms are combined.

Consider this in a real-world context: if you have a group of items represented by 2a + 6 and you remove a group represented by 5b + 2, you are left with the difference between these groups. The simplification process helps to express this difference in its simplest form.

In summary, to simplify (2a + 6) - (5b + 2), we distribute the negative sign and combine like terms: 2a + 6 - 5b - 2. Combining the constants gives us 2a - 5b + 4. This is the simplified form of the expression.

9. -m⁴ + m² - m³ + 5m² + 2m⁴

To find the sum of the monomials -m⁴, m², -m³, 5m², and 2m⁴, we need to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two sets of like terms: terms with m⁴ and terms with m².

First, let's identify the terms with m⁴: -m⁴ and 2m⁴. These are like terms because they both have the variable 'm' raised to the power of 4. We can add their coefficients:

-1 + 2 = 1

So, the sum of these terms is 1m⁴, which is simply m⁴.

Next, let's identify the terms with m²: m² and 5m². These are like terms because they both have the variable 'm' raised to the power of 2. We can add their coefficients:

1 + 5 = 6

So, the sum of these terms is 6m².

Now, we have the term -m³, which does not have any like terms in the expression. It remains as is.

Combine the sums of the like terms and the remaining term:

m⁴ + 6m² - m³

It is conventional to write polynomials in descending order of the exponents, so we can rearrange the terms:

m⁴ - m³ + 6m²

This is the simplified form of the expression. This example demonstrates the process of combining like terms in a polynomial expression with multiple terms and different powers of the variable.

Consider this in a context of algebraic simplification: we are essentially grouping together terms that can be combined based on their degree (the exponent of the variable). Terms with the same degree can be added or subtracted, while terms with different degrees remain separate.

In summary, to find the sum of -m⁴ + m² - m³ + 5m² + 2m⁴, we combine like terms: -m⁴ + 2m⁴ = m⁴ and m² + 5m² = 6m². The term -m³ remains unchanged. The final expression, written in descending order of exponents, is m⁴ - m³ + 6m².

10. 2n² - 4n - 3n² - 2n

To find the sum of the monomials 2n², -4n, -3n², and -2n, we need to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two sets of like terms: terms with n² and terms with n.

First, let's identify the terms with n²: 2n² and -3n². These are like terms because they both have the variable 'n' raised to the power of 2. We can add their coefficients:

2 + (-3) = -1

So, the sum of these terms is -1n², which is simply -n².

Next, let's identify the terms with n: -4n and -2n. These are like terms because they both have the variable 'n' raised to the power of 1. We can add their coefficients:

-4 + (-2) = -6

So, the sum of these terms is -6n.

Now, we combine the sums of the like terms:

-n² - 6n

This is the simplified form of the expression. This example demonstrates the process of combining like terms in an algebraic expression, including handling negative coefficients.

Consider this in the context of simplifying polynomials: we are grouping terms that have the same degree (the exponent of the variable) and combining their coefficients. This process makes the expression simpler and easier to work with.

In summary, to find the sum of 2n² - 4n - 3n² - 2n, we combine like terms: 2n² + (-3n²) = -n² and -4n + (-2n) = -6n. The final expression is -n² - 6n.

Finding the sum of monomials involves identifying and combining like terms. This fundamental algebraic skill is crucial for simplifying expressions and solving equations. By understanding the principles of combining like terms, you can effectively manipulate algebraic expressions and solve a wide range of mathematical problems. This guide has provided detailed explanations and examples to help you master this essential concept in mathematics.