Factorize the numerator and denominator

Oct 13, 2025 by thesweetdesigns.com 62 views

Introduction

In mathematics, algebraic expressions are used to represent various mathematical operations. When dealing with algebraic expressions, it is essential to follow a specific set of steps to simplify and solve them. In this article, we will determine the steps used to find the product of two given algebraic expressions.

Given Expression

The given expression is:

\frac{x^2+7x+10}{x^2+6x+4} \cdot \frac{z^2+3x+2}{x^1+6z+5}

Possible Steps

To find the product of the given expressions, we need to follow a specific set of steps. The possible steps are:

Factorize the numerator and denominator: Factorize the numerator and denominator of each expression to simplify them.
Cancel out common factors: Cancel out any common factors between the numerator and denominator of each expression.
Multiply the expressions: Multiply the simplified expressions to find the product.
Simplify the product: Simplify the product by combining like terms.

Step-by-Step Solution

Step 1: Factorize the Numerator and Denominator

To factorize the numerator and denominator, we need to find the greatest common factor (GCF) of each expression.

For the first expression, the numerator is $x^2+7x+10$ and the denominator is $x^2+6x+4$ . The GCF of the numerator is $(x+5)$ and the GCF of the denominator is $(x+2)$ .
For the second expression, the numerator is $z^2+3x+2$ and the denominator is $x^1+6z+5$ . The GCF of the numerator is $(z+1)$ and the GCF of the denominator is $(x+5)$ .

# Step 1: Factorize the Numerator and Denominator

## First Expression

*   Numerator: $x^2+7x+10$
*   Denominator: $x^2+6x+4$
*   GCF of Numerator: $(x+5)$
*   GCF of Denominator: $(x+2)$

## Second Expression

*   Numerator: $z^2+3x+2$
*   Denominator: $x^1+6z+5$
*   GCF of Numerator: $(z+1)$
*   GCF of Denominator: $(x+5)$

Step 2: Cancel Out Common Factors

After factorizing the numerator and denominator, we can cancel out any common factors between the numerator and denominator of each expression.

For the first expression, we can cancel out the common factor $(x+2)$ between the numerator and denominator.
For the second expression, we can cancel out the common factor $(x+5)$ between the numerator and denominator.

# Step 2: Cancel Out Common Factors

## First Expression

*   Cancel out $(x+2)$ between the numerator and denominator
*   Simplified Expression: $\frac{(x+5)}{1}$

## Second Expression

*   Cancel out $(x+5)$ between the numerator and denominator
*   Simplified Expression: $\frac{(z+1)}{1}$

Step 3: Multiply the Expressions

After canceling out common factors, we can multiply the simplified expressions to find the product.

The product of the two simplified expressions is $(x+5) \cdot (z+1)$ .

# Step 3: Multiply the Expressions

## Product of Simplified Expressions

*   $(x+5) \cdot (z+1)$

Step 4: Simplify the Product

After multiplying the expressions, we can simplify the product by combining like terms.

The simplified product is $xz + x + 5z + 5$ .

# Step 4: Simplify the Product

## Simplified Product

*   $xz + x + 5z + 5$

Conclusion

In this article, we determined the steps used to find the product of two given algebraic expressions. The steps involved factorizing the numerator and denominator, canceling out common factors, multiplying the expressions, and simplifying the product. By following these steps, we were able to find the product of the given expressions.

Final Answer

$The final answer is $\boxed{xz + x + 5z + 5}$.<br/> **Determine which steps are used to find the product shown: Q&A**$

Introduction

In our previous article, we determined the steps used to find the product of two given algebraic expressions. In this article, we will provide a Q&A section to further clarify the steps involved in finding the product of algebraic expressions.

Q&A

Q: What is the first step in finding the product of two algebraic expressions?

A: The first step in finding the product of two algebraic expressions is to factorize the numerator and denominator of each expression.

Q: How do I factorize the numerator and denominator?

A: To factorize the numerator and denominator, you need to find the greatest common factor (GCF) of each expression. The GCF is the largest factor that divides both the numerator and denominator without leaving a remainder.

Q: What is the next step after factorizing the numerator and denominator?

A: After factorizing the numerator and denominator, you need to cancel out any common factors between the numerator and denominator of each expression.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to identify the common factors between the numerator and denominator of each expression and cancel them out.

Q: What is the next step after canceling out common factors?

A: After canceling out common factors, you need to multiply the simplified expressions to find the product.

Q: How do I multiply the simplified expressions?

A: To multiply the simplified expressions, you need to multiply the numerators and denominators separately and then simplify the result.

Q: What is the final step in finding the product of two algebraic expressions?

A: The final step in finding the product of two algebraic expressions is to simplify the product by combining like terms.

Q: How do I simplify the product?

A: To simplify the product, you need to combine like terms by adding or subtracting the coefficients of the same variables.

Example Questions

Q: Find the product of the following expressions:

\frac{x^2+5x+6}{x^2+4x+4} \cdot \frac{y^2+3y+2}{y^2+5y+5}

A: To find the product of the given expressions, we need to follow the steps outlined above.

Factorize the numerator and denominator: $\frac{(x+2)(x+3)}{(x+2)^2} \cdot \frac{(y+1)(y+2)}{(y+1)(y+5)}$
Cancel out common factors: $\frac{(x+3)}{(x+2)} \cdot \frac{(y+2)}{(y+5)}$
Multiply the expressions: $(x+3)(y+2)$
Simplify the product: $xy + 2x + 3y + 6$

Q: Find the product of the following expressions:

\frac{x^2+3x+2}{x^2+5x+4} \cdot \frac{z^2+2z+1}{z^2+4z+3}

A: To find the product of the given expressions, we need to follow the steps outlined above.

Factorize the numerator and denominator: $\frac{(x+1)(x+2)}{(x+1)(x+4)} \cdot \frac{(z+1)^2}{(z+1)(z+3)}$
Cancel out common factors: $\frac{(x+2)}{(x+4)} \cdot \frac{(z+1)}{(z+3)}$
Multiply the expressions: $(x+2)(z+1)$
Simplify the product: $xz + x + 2z + 2$

Conclusion

In this article, we provided a Q&A section to further clarify the steps involved in finding the product of algebraic expressions. We also provided example questions to demonstrate how to apply the steps outlined above. By following these steps, you can find the product of any two algebraic expressions.

Final Answer

The final answer is $\boxed{xy + 2x + 3y + 6}$ and $\boxed{xz + x + 2z + 2}$ .