Importance of Multiplying Expressions

Oct 13, 2025 by miodeya.com 45 views

Introduction

In algebra, multiplying expressions is a fundamental operation that helps us simplify complex equations and solve problems. In this article, we will focus on multiplying two algebraic expressions, and we will use the given problem as a case study. We will also discuss the importance of multiplying expressions in mathematics and provide a step-by-step guide on how to do it.

The Problem

The problem we are given is to multiply the following expressions:

{ \frac{3x^2 + 2x - 21}{-2x^2 - 2x + 12} \cdot \frac{2x^2 + 25x + 63}{6x^3 + 7x - 49} \}

Step 1: Factorize the Expressions

To multiply the expressions, we need to factorize them first. Factorizing an expression means breaking it down into simpler expressions that can be multiplied together.

Let's start with the first expression:

{ \frac{3x^2 + 2x - 21}{-2x^2 - 2x + 12} \}

We can factorize the numerator and denominator separately:

Numerator:

$3x^2 + 2x - 21 = (3x - 7)(x + 3)$

Denominator:

$-2x^2 - 2x + 12 = -(2x - 3)(x + 4)$

So, the first expression can be written as:

{ \frac{(3x - 7)(x + 3)}{-(2x - 3)(x + 4)} \}

Now, let's factorize the second expression:

{ \frac{2x^2 + 25x + 63}{6x^3 + 7x - 49} \}

We can factorize the numerator and denominator separately:

Numerator:

$2x^2 + 25x + 63 = (2x + 9)(x + 7)$

Denominator:

$6x^3 + 7x - 49 = (x + 7)(2x - 1)(3x + 7)$

So, the second expression can be written as:

{ \frac{(2x + 9)(x + 7)}{(x + 7)(2x - 1)(3x + 7)} \}

Step 2: Cancel Out Common Factors

Now that we have factorized both expressions, we can cancel out common factors. In this case, we can cancel out the $(x + 7)$ term from the numerator and denominator of both expressions.

So, the first expression becomes:

{ \frac{(3x - 7)}{-(2x - 3)} \}

And the second expression becomes:

{ \frac{(2x + 9)}{(2x - 1)(3x + 7)} \}

Step 3: Multiply the Expressions

Now that we have simplified both expressions, we can multiply them together.

{ \frac{(3x - 7)}{-(2x - 3)} \cdot \frac{(2x + 9)}{(2x - 1)(3x + 7)} \}

To multiply these expressions, we need to multiply the numerators and denominators separately.

Numerators:

$(3x - 7)(2x + 9) = 6x^2 + 27x - 14x - 63 = 6x^2 + 13x - 63$

Denominators:

$-(2x - 3)(2x - 1)(3x + 7) = -(4x^2 - 8x + 3)(3x + 7) = -(12x^3 + 28x^2 - 21x - 21)$

So, the product of the two expressions is:

{ \frac{6x^2 + 13x - 63}{-(12x^3 + 28x^2 - 21x - 21)} \}

Conclusion

In this article, we have multiplied two algebraic expressions using the given problem as a case study. We have factorized the expressions, canceled out common factors, and multiplied the expressions together. The final product is a simplified expression that can be used to solve problems in mathematics.

Importance of Multiplying Expressions

Multiplying expressions is an important operation in mathematics that helps us simplify complex equations and solve problems. It is used in various fields such as algebra, calculus, and physics to solve problems and make predictions.

Real-World Applications

Multiplying expressions has many real-world applications. For example, it is used in engineering to design and optimize systems, in economics to model and analyze economic systems, and in physics to describe and predict the behavior of physical systems.

Tips and Tricks

Here are some tips and tricks to help you multiply expressions:

Factorize the expressions before multiplying them.
Cancel out common factors to simplify the expressions.
Multiply the numerators and denominators separately.
Use algebraic identities to simplify the expressions.

Conclusion

Introduction

In our previous article, we discussed how to multiply algebraic expressions using the given problem as a case study. We factorized the expressions, canceled out common factors, and multiplied the expressions together. In this article, we will provide a Q&A guide to help you understand the concept of multiplying algebraic expressions.

Q: What is the purpose of multiplying algebraic expressions?

A: The purpose of multiplying algebraic expressions is to simplify complex equations and solve problems in mathematics. It is used in various fields such as algebra, calculus, and physics to solve problems and make predictions.

Q: How do I multiply algebraic expressions?

A: To multiply algebraic expressions, you need to follow these steps:

Factorize the expressions before multiplying them.
Cancel out common factors to simplify the expressions.
Multiply the numerators and denominators separately.
Use algebraic identities to simplify the expressions.

Q: What are some common algebraic identities that I can use to simplify expressions?

A: Some common algebraic identities that you can use to simplify expressions include:

The distributive property: a(b + c) = ab + ac
The commutative property: a + b = b + a
The associative property: (a + b) + c = a + (b + c)
The identity property: a + 0 = a
The inverse property: a + (-a) = 0

Q: How do I factorize algebraic expressions?

A: To factorize algebraic expressions, you need to find the greatest common factor (GCF) of the terms and divide each term by the GCF. You can also use the following methods to factorize expressions:

Factoring out a common term: If all the terms have a common factor, you can factor it out.
Factoring by grouping: If the terms can be grouped into pairs, you can factor out a common factor from each pair.
Factoring using the difference of squares: If the expression is in the form of a^2 - b^2, you can factor it as (a + b)(a - b).

Q: What are some common mistakes to avoid when multiplying algebraic expressions?

A: Some common mistakes to avoid when multiplying algebraic expressions include:

Not factorizing the expressions before multiplying them.
Not canceling out common factors.
Not multiplying the numerators and denominators separately.
Not using algebraic identities to simplify the expressions.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you need to follow these steps:

Factorize the expressions.
Cancel out common factors.
Multiply the numerators and denominators separately.
Use algebraic identities to simplify the expressions.

Q: What are some real-world applications of multiplying algebraic expressions?

A: Some real-world applications of multiplying algebraic expressions include:

Engineering: Multiplying algebraic expressions is used to design and optimize systems.
Economics: Multiplying algebraic expressions is used to model and analyze economic systems.
Physics: Multiplying algebraic expressions is used to describe and predict the behavior of physical systems.

Conclusion

In conclusion, multiplying algebraic expressions is an important operation in mathematics that helps us simplify complex equations and solve problems. By following the steps outlined in this article, you can multiply algebraic expressions and solve problems in mathematics. Remember to factorize the expressions, cancel out common factors, multiply the numerators and denominators separately, and use algebraic identities to simplify the expressions.