Factor The Following Expression Completely:$\[ 28z^{19} + 14z^{13} + 21z^7 = \\]$\[ \square \\]

Feb 25, 2025 by ADMIN 96 views

**Factoring the Given Expression Completely**

=====================================================

Introduction

Factoring expressions is a crucial concept in algebra, and it plays a vital role in solving various mathematical problems. In this article, we will focus on factoring the given expression completely. The expression we need to factor is $28z^{19} + 14z^{13} + 21z^7$ . Factoring this expression will help us simplify it and make it easier to work with.

Understanding the Expression

Before we start factoring the expression, let's understand what it represents. The expression $28z^{19} + 14z^{13} + 21z^7$ is a polynomial expression, where $z$ is the variable. The coefficients of the expression are $28$ , $14$ , and $21$ , and the powers of $z$ are $19$ , $13$ , and $7$ , respectively.

Factoring Out the Greatest Common Factor (GCF)

The first step in factoring the expression is to find the greatest common factor (GCF) of the three terms. The GCF is the largest factor that divides all three terms without leaving a remainder. In this case, the GCF of $28z^{19}$ , $14z^{13}$ , and $21z^7$ is $7z^7$ . We can factor out the GCF from each term as follows:

28z^{19} + 14z^{13} + 21z^7 = 7z^7(4z^{12} + 2z^6 + 3)

Factoring the Quadratic Expression

The expression inside the parentheses, $4z^{12} + 2z^6 + 3$ , is a quadratic expression. We can factor this expression by finding two numbers whose product is $12$ and whose sum is $2$ . However, this expression cannot be factored further using simple factoring techniques.

Factoring the Expression Completely

Since we cannot factor the quadratic expression further, we can conclude that the expression $28z^{19} + 14z^{13} + 21z^7$ is factored completely as follows:

28z^{19} + 14z^{13} + 21z^7 = 7z^7(4z^{12} + 2z^6 + 3)

Conclusion

In this article, we factored the given expression completely. We started by finding the greatest common factor (GCF) of the three terms and factored it out. Then, we tried to factor the quadratic expression inside the parentheses, but it cannot be factored further using simple factoring techniques. Therefore, we concluded that the expression $28z^{19} + 14z^{13} + 21z^7$ is factored completely as $7z^7(4z^{12} + 2z^6 + 3)$ .

Example Problems

Problem 1

Factor the expression $24x^5 + 12x^3 + 18x$ completely.

Solution

The GCF of $24x^5$ , $12x^3$ , and $18x$ is $6x$ . We can factor it out as follows:

24x^5 + 12x^3 + 18x = 6x(4x^4 + 2x^2 + 3)

Since the quadratic expression inside the parentheses cannot be factored further, we conclude that the expression is factored completely as $6x(4x^4 + 2x^2 + 3)$ .

Problem 2

Factor the expression $36y^9 + 18y^6 + 27y^3$ completely.

Solution

The GCF of $36y^9$ , $18y^6$ , and $27y^3$ is $9y^3$ . We can factor it out as follows:

36y^9 + 18y^6 + 27y^3 = 9y^3(4y^6 + 2y^3 + 3)

Since the quadratic expression inside the parentheses cannot be factored further, we conclude that the expression is factored completely as $9y^3(4y^6 + 2y^3 + 3)$ .

Tips and Tricks

When factoring an expression, start by finding the greatest common factor (GCF) of the terms.
If the GCF is a constant, factor it out from each term.
If the GCF is a variable, factor it out from each term along with its power.
If the expression inside the parentheses is a quadratic expression, try to factor it by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
If the quadratic expression cannot be factored further, leave it as is.

Common Mistakes

Not finding the greatest common factor (GCF) of the terms before factoring.
Factoring out a term that is not the greatest common factor (GCF) of the terms.
Not checking if the quadratic expression inside the parentheses can be factored further.
Leaving the quadratic expression inside the parentheses as is without trying to factor it.

Real-World Applications

Factoring expressions has numerous real-world applications in various fields, including:

Algebra: Factoring expressions is a crucial concept in algebra, and it plays a vital role in solving various mathematical problems.
Calculus: Factoring expressions is used in calculus to simplify complex expressions and make them easier to work with.
Physics: Factoring expressions is used in physics to simplify complex equations and make them easier to solve.
Engineering: Factoring expressions is used in engineering to simplify complex equations and make them easier to solve.

Conclusion

In conclusion, factoring expressions is a crucial concept in algebra, and it plays a vital role in solving various mathematical problems. We started by factoring the given expression completely and then provided example problems and tips and tricks for factoring expressions. We also discussed common mistakes to avoid and real-world applications of factoring expressions. By following the steps outlined in this article, you can master the art of factoring expressions and become proficient in algebra.

=============================

Introduction

Factoring expressions is a crucial concept in algebra, and it plays a vital role in solving various mathematical problems. In our previous article, we discussed how to factor expressions completely and provided example problems and tips and tricks for factoring expressions. In this article, we will answer some frequently asked questions about factoring expressions.

Q&A

Q: What is factoring?

A: Factoring is the process of expressing an expression as a product of simpler expressions, called factors.

Q: Why is factoring important?

A: Factoring is important because it helps us simplify complex expressions and make them easier to work with. It also helps us identify the underlying structure of an expression and make it easier to solve.

Q: How do I factor an expression?

A: To factor an expression, start by finding the greatest common factor (GCF) of the terms. If the GCF is a constant, factor it out from each term. If the GCF is a variable, factor it out from each term along with its power. If the expression inside the parentheses is a quadratic expression, try to factor it by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides all the terms of an expression without leaving a remainder.

Q: How do I find the GCF of an expression?

A: To find the GCF of an expression, list all the factors of each term and find the greatest common factor among them.

Q: Can I factor an expression that has a variable in the denominator?

A: Yes, you can factor an expression that has a variable in the denominator. However, you need to be careful when factoring out the GCF, as it may affect the denominator.

Q: Can I factor an expression that has a negative sign in front of it?

A: Yes, you can factor an expression that has a negative sign in front of it. However, you need to be careful when factoring out the GCF, as it may affect the sign of the expression.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, try to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. If you cannot find such numbers, the expression cannot be factored further.

Q: Can I factor an expression that has a variable in the exponent?

A: Yes, you can factor an expression that has a variable in the exponent. However, you need to be careful when factoring out the GCF, as it may affect the exponent.

Q: Can I factor an expression that has a fraction in it?

A: Yes, you can factor an expression that has a fraction in it. However, you need to be careful when factoring out the GCF, as it may affect the fraction.

Tips and Tricks

When factoring an expression, start by finding the greatest common factor (GCF) of the terms.
If the GCF is a constant, factor it out from each term.
If the GCF is a variable, factor it out from each term along with its power.
If the expression inside the parentheses is a quadratic expression, try to factor it by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Be careful when factoring out the GCF, as it may affect the sign of the expression or the exponent.

Common Mistakes

Not finding the greatest common factor (GCF) of the terms before factoring.
Factoring out a term that is not the greatest common factor (GCF) of the terms.
Not checking if the quadratic expression inside the parentheses can be factored further.
Leaving the quadratic expression inside the parentheses as is without trying to factor it.

Real-World Applications

Factoring expressions has numerous real-world applications in various fields, including:

Algebra: Factoring expressions is a crucial concept in algebra, and it plays a vital role in solving various mathematical problems.
Calculus: Factoring expressions is used in calculus to simplify complex expressions and make them easier to work with.
Physics: Factoring expressions is used in physics to simplify complex equations and make them easier to solve.
Engineering: Factoring expressions is used in engineering to simplify complex equations and make them easier to solve.

Conclusion

In conclusion, factoring expressions is a crucial concept in algebra, and it plays a vital role in solving various mathematical problems. We answered some frequently asked questions about factoring expressions and provided tips and tricks for factoring expressions. By following the steps outlined in this article, you can master the art of factoring expressions and become proficient in algebra.