Add The Following Expressions:(i) ${ 5m(3-m) }$ And ${ 6m^2 - 13m }$(ii) ${ 4y(3y^2 + 5y - 7) }$ And ${ 2(y^3 - 4y^2 + 5) }$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and expanding and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of expanding and simplifying two given algebraic expressions, and provide a step-by-step guide on how to tackle similar problems.

Expanding Algebraic Expressions

Expanding algebraic expressions involves multiplying the terms within the expression to obtain a simplified form. This process can be achieved using the distributive property, which states that for any real numbers a, b, and c:

a(b + c) = ab + ac

Using this property, we can expand the given expressions.

(i) Expanding $5m(3-m)$

To expand the expression $5m(3-m)$, we will use the distributive property to multiply the terms within the parentheses.

5m(3-m) = 5m(3) - 5m(m)

Now, we can simplify the expression by multiplying the terms.

5m(3) = 15m
5m(m) = 5m^2

Therefore, the expanded form of the expression $5m(3-m)$ is:

15m - 5m^2

(ii) Expanding $6m^2 - 13m$

The expression $6m^2 - 13m$ is already in its simplest form, as it is a polynomial expression with no terms to expand.

(iii) Expanding $4y(3y^2 + 5y - 7)$

To expand the expression $4y(3y^2 + 5y - 7)$, we will use the distributive property to multiply the terms within the parentheses.

4y(3y^2 + 5y - 7) = 4y(3y^2) + 4y(5y) - 4y(7)

Now, we can simplify the expression by multiplying the terms.

4y(3y^2) = 12y^3
4y(5y) = 20y^2
4y(7) = 28y

Therefore, the expanded form of the expression $4y(3y^2 + 5y - 7)$ is:

12y^3 + 20y^2 - 28y

(iv) Expanding $2(y^3 - 4y^2 + 5)$

To expand the expression $2(y^3 - 4y^2 + 5)$, we will use the distributive property to multiply the terms within the parentheses.

2(y^3 - 4y^2 + 5) = 2(y^3) - 2(4y^2) + 2(5)

Now, we can simplify the expression by multiplying the terms.

2(y^3) = 2y^3
2(4y^2) = 8y^2
2(5) = 10

Therefore, the expanded form of the expression $2(y^3 - 4y^2 + 5)$ is:

2y^3 - 8y^2 + 10

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms to obtain a simplified form. This process can be achieved by adding or subtracting the coefficients of the like terms.

(i) Simplifying $15m - 5m^2$

To simplify the expression $15m - 5m^2$, we will combine the like terms.

15m - 5m^2 = -5m^2 + 15m

(ii) Simplifying $12y^3 + 20y^2 - 28y$

To simplify the expression $12y^3 + 20y^2 - 28y$, we will combine the like terms.

12y^3 + 20y^2 - 28y = 12y^3 + 20y^2 - 28y

(iii) Simplifying $2y^3 - 8y^2 + 10$

To simplify the expression $2y^3 - 8y^2 + 10$, we will combine the like terms.

2y^3 - 8y^2 + 10 = 2y^3 - 8y^2 + 10

Conclusion

Expanding and simplifying algebraic expressions is a crucial skill for students and professionals alike. By using the distributive property and combining like terms, we can simplify complex expressions and obtain a more manageable form. In this article, we have explored the process of expanding and simplifying two given algebraic expressions, and provided a step-by-step guide on how to tackle similar problems.

Final Thoughts

Q: What is the distributive property, and how is it used to expand algebraic expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property is used to expand algebraic expressions by multiplying the terms within the parentheses.

Q: How do I expand an algebraic expression with multiple terms within the parentheses?

A: To expand an algebraic expression with multiple terms within the parentheses, you can use the distributive property to multiply each term within the parentheses by the term outside the parentheses.

For example, to expand the expression $4y(3y^2 + 5y - 7)$, you would multiply each term within the parentheses by 4y:

4y(3y^2 + 5y - 7) = 4y(3y^2) + 4y(5y) - 4y(7)

Q: How do I simplify an algebraic expression with like terms?

A: To simplify an algebraic expression with like terms, you can combine the coefficients of the like terms.

For example, to simplify the expression $15m - 5m^2$, you would combine the like terms:

15m - 5m^2 = -5m^2 + 15m

Q: What is the difference between expanding and simplifying an algebraic expression?

A: Expanding an algebraic expression involves multiplying the terms within the expression to obtain a more complex form. Simplifying an algebraic expression involves combining like terms to obtain a simpler form.

For example, the expression $4y(3y^2 + 5y - 7)$ is expanded to:

12y^3 + 20y^2 - 28y

And the expression $15m - 5m^2$ is simplified to:

-5m^2 + 15m

Q: How do I know when to expand or simplify an algebraic expression?

A: You should expand an algebraic expression when you need to multiply the terms within the expression to obtain a more complex form. You should simplify an algebraic expression when you need to combine like terms to obtain a simpler form.

For example, if you are given the expression $4y(3y^2 + 5y - 7)$, you would expand it to:

12y^3 + 20y^2 - 28y

But if you are given the expression $15m - 5m^2$, you would simplify it to:

-5m^2 + 15m

Q: What are some common mistakes to avoid when expanding and simplifying algebraic expressions?

A: Some common mistakes to avoid when expanding and simplifying algebraic expressions include:

• Not using the distributive property to expand expressions
• Not combining like terms when simplifying expressions
• Not checking for errors in the expansion or simplification process

Q: How can I practice expanding and simplifying algebraic expressions?

A: You can practice expanding and simplifying algebraic expressions by working through examples and exercises in a textbook or online resource. You can also try creating your own examples and exercises to practice your skills.

Conclusion

Expanding and simplifying algebraic expressions is a crucial skill for students and professionals alike. By mastering this skill, you can tackle complex problems and obtain a deeper understanding of mathematical concepts. Remember to use the distributive property to expand expressions, combine like terms to simplify expressions, and check for errors in the expansion or simplification process. With practice and patience, you can become proficient in expanding and simplifying algebraic expressions.