Algebraic Expressions: ProductsExpand:${ 3t 2(3t 3+5) + 5t 3(4t 2+4) } A N S W E R : Answer: A N S W Er : { \square \}

Mar 8, 2025 by ADMIN 120 views

**Algebraic Expressions: Products and Expansion**

Introduction

Algebraic expressions are a fundamental concept in mathematics, and understanding how to expand and simplify them is crucial for solving various mathematical problems. In this article, we will focus on expanding algebraic expressions involving products, which is a critical skill for students to master. We will use the given expression $3t^2(3t^3+5) + 5t^3(4t^2+4)$ as an example to demonstrate the step-by-step process of expanding products.

What are Algebraic Expressions?

Algebraic expressions are a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division. They can be represented using variables, numbers, and mathematical symbols. Algebraic expressions can be simple or complex, and they can be used to represent real-world situations or mathematical concepts.

Types of Algebraic Expressions

There are several types of algebraic expressions, including:

Monomials: A monomial is a single term that consists of a variable or a constant. For example, $3x$ is a monomial.
Binomials: A binomial is a two-term expression that consists of two variables or constants. For example, $x + 3$ is a binomial.
Polynomials: A polynomial is a multi-term expression that consists of two or more variables or constants. For example, $x + 3 + 2x^2$ is a polynomial.

Expanding Algebraic Expressions

Expanding algebraic expressions involves multiplying the terms together to simplify the expression. There are several rules to follow when expanding algebraic expressions:

Distributive Property: The distributive property states that for any numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ .
Commutative Property: The commutative property states that for any numbers $a$ and $b$ , $ab = ba$ .
Associative Property: The associative property states that for any numbers $a$ , $b$ , and $c$ , $(ab)c = a(bc)$ .

Step-by-Step Process of Expanding Products

To expand the given expression $3t^2(3t^3+5) + 5t^3(4t^2+4)$ , we will follow the step-by-step process outlined below:

Step 1: Distribute the First Term

The first term is $3t^2(3t^3+5)$ . We will distribute the $3t^2$ to both terms inside the parentheses:

3t^2(3t^3+5) = 3t^2 \cdot 3t^3 + 3t^2 \cdot 5

Step 2: Simplify the First Term

We will simplify the first term by multiplying the variables and constants:

3t^2 \cdot 3t^3 = 9t^5

3t^2 \cdot 5 = 15t^2

So, the first term becomes:

3t^2(3t^3+5) = 9t^5 + 15t^2

Step 3: Distribute the Second Term

The second term is $5t^3(4t^2+4)$ . We will distribute the $5t^3$ to both terms inside the parentheses:

5t^3(4t^2+4) = 5t^3 \cdot 4t^2 + 5t^3 \cdot 4

Step 4: Simplify the Second Term

We will simplify the second term by multiplying the variables and constants:

5t^3 \cdot 4t^2 = 20t^5

5t^3 \cdot 4 = 20t^3

So, the second term becomes:

5t^3(4t^2+4) = 20t^5 + 20t^3

Step 5: Combine Like Terms

We will combine the like terms from both expressions:

9t^5 + 15t^2 + 20t^5 + 20t^3

We can combine the $t^5$ terms:

9t^5 + 20t^5 = 29t^5

So, the final expression becomes:

29t^5 + 15t^2 + 20t^3

Conclusion

Expanding algebraic expressions involving products requires a step-by-step process that involves distributing the terms, simplifying the expression, and combining like terms. By following the distributive property, commutative property, and associative property, we can simplify complex algebraic expressions and solve mathematical problems. In this article, we used the given expression $3t^2(3t^3+5) + 5t^3(4t^2+4)$ as an example to demonstrate the step-by-step process of expanding products.

Practice Problems

To practice expanding algebraic expressions, try the following problems:

Expand the expression $2x(3x^2+4) + 3x(2x^2+5)$ .
Expand the expression $4t^2(3t^3+2) + 2t^3(5t^2+3)$ .
Expand the expression $x(2x^2+3) + 2x(4x^2+5)$ .

Answer Key

$6x^3 + 8x + 6x^3 + 15x$
$12t^5 + 8t^2 + 10t^5 + 6t^3$
$2x^3 + 3x + 8x^3 + 10x$

References

Algebraic Expressions: Khan Academy. (n.d.). Algebraic Expressions. Retrieved from https://www.khanacademy.org/math/algebra
Distributive Property: Math Open Reference. (n.d.). Distributive Property. Retrieved from https://www.mathopenref.com/distributive.html
Commutative Property: Math Open Reference. (n.d.). Commutative Property. Retrieved from https://www.mathopenref.com/commutative.html
Associative Property: Math Open Reference. (n.d.). Associative Property. Retrieved from https://www.mathopenref.com/associative.html
Algebraic Expressions: Products and Expansion Q&A =====================================================

Introduction

In our previous article, we discussed the concept of algebraic expressions and how to expand them using the distributive property, commutative property, and associative property. In this article, we will provide a Q&A section to help students and teachers clarify any doubts they may have about expanding algebraic expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical rule that states that for any numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . This means that we can distribute a single term to both terms inside the parentheses.

Q: How do I apply the distributive property to expand an algebraic expression?

A: To apply the distributive property, simply multiply the single term to both terms inside the parentheses. For example, if we have the expression $3x(2x^2+4)$ , we would multiply the $3x$ to both terms inside the parentheses: $3x \cdot 2x^2 + 3x \cdot 4$ .

Q: What is the commutative property?

A: The commutative property is a mathematical rule that states that for any numbers $a$ and $b$ , $ab = ba$ . This means that we can change the order of the terms without changing the value of the expression.

Q: How do I apply the commutative property to expand an algebraic expression?

A: To apply the commutative property, simply change the order of the terms. For example, if we have the expression $2x^2 + 3x$ , we can change the order of the terms to get $3x + 2x^2$ .

Q: What is the associative property?

A: The associative property is a mathematical rule that states that for any numbers $a$ , $b$ , and $c$ , $(ab)c = a(bc)$ . This means that we can change the order of the terms without changing the value of the expression.

Q: How do I apply the associative property to expand an algebraic expression?

A: To apply the associative property, simply change the order of the terms. For example, if we have the expression $(2x^2 + 3x)(4x^2 + 5)$ , we can change the order of the terms to get $(3x + 2x^2)(4x^2 + 5)$ .

Q: How do I combine like terms when expanding an algebraic expression?

A: To combine like terms, simply add or subtract the coefficients of the like terms. For example, if we have the expression $2x^2 + 3x^2$ , we can combine the like terms to get $5x^2$ .

Q: What is the difference between a monomial and a polynomial?

A: A monomial is a single term that consists of a variable or a constant. For example, $3x$ is a monomial. A polynomial is a multi-term expression that consists of two or more variables or constants. For example, $x + 3 + 2x^2$ is a polynomial.

Q: How do I expand a polynomial expression?

A: To expand a polynomial expression, simply apply the distributive property, commutative property, and associative property to each term in the expression. For example, if we have the expression $2x^2(3x^3+4) + 3x(2x^2+5)$ , we would apply the distributive property to each term to get $6x^5 + 8x^2 + 6x^3 + 15x$ .