4. Factor Each Of The Following Expressions.(a) \[$5x^2 + X\$\](b) \[$a^2 + 3a\$\](c) \[$5n^2 + 2n\$\](d) \[$6n^2 + 3n\$\](e) \[$5n^2 - 10n\$\](f) \[$3x^2 + 6x\$\](g) \[$15x^2 + 30x\$\](h)

In mathematics, factoring is a process of expressing an algebraic expression as a product of simpler expressions. It is an essential concept in algebra and is used to simplify complex expressions, solve equations, and find the roots of polynomials. In this article, we will factor each of the given expressions using various factoring techniques.

(a) Factor the expression $5x^2 + x$

To factor the expression $5x^2 + x$, we need to find the greatest common factor (GCF) of the two terms. The GCF of $5x^2$ and $x$ is $x$. We can factor out the GCF as follows:

$$5x^2 + x = x(5x + 1)$$

Therefore, the factored form of the expression $5x^2 + x$ is $x(5x + 1)$.

(b) Factor the expression $a^2 + 3a$

To factor the expression $a^2 + 3a$, we need to find the GCF of the two terms. The GCF of $a^2$ and $3a$ is $a$. We can factor out the GCF as follows:

$$a^2 + 3a = a(a + 3)$$

Therefore, the factored form of the expression $a^2 + 3a$ is $a(a + 3)$.

(c) Factor the expression $5n^2 + 2n$

To factor the expression $5n^2 + 2n$, we need to find the GCF of the two terms. The GCF of $5n^2$ and $2n$ is $n$. We can factor out the GCF as follows:

$$5n^2 + 2n = n(5n + 2)$$

Therefore, the factored form of the expression $5n^2 + 2n$ is $n(5n + 2)$.

(d) Factor the expression $6n^2 + 3n$

To factor the expression $6n^2 + 3n$, we need to find the GCF of the two terms. The GCF of $6n^2$ and $3n$ is $3n$. We can factor out the GCF as follows:

$$6n^2 + 3n = 3n(2n + 1)$$

Therefore, the factored form of the expression $6n^2 + 3n$ is $3n(2n + 1)$.

(e) Factor the expression $5n^2 - 10n$

To factor the expression $5n^2 - 10n$, we need to find the GCF of the two terms. The GCF of $5n^2$ and $10n$ is $5n$. We can factor out the GCF as follows:

$$5n^2 - 10n = 5n(n - 2)$$

Therefore, the factored form of the expression $5n^2 - 10n$ is $5n(n - 2)$.

(f) Factor the expression $3x^2 + 6x$

To factor the expression $3x^2 + 6x$, we need to find the GCF of the two terms. The GCF of $3x^2$ and $6x$ is $3x$. We can factor out the GCF as follows:

$$3x^2 + 6x = 3x(x + 2)$$

Therefore, the factored form of the expression $3x^2 + 6x$ is $3x(x + 2)$.

(g) Factor the expression $15x^2 + 30x$

To factor the expression $15x^2 + 30x$, we need to find the GCF of the two terms. The GCF of $15x^2$ and $30x$ is $15x$. We can factor out the GCF as follows:

$$15x^2 + 30x = 15x(x + 2)$$

Therefore, the factored form of the expression $15x^2 + 30x$ is $15x(x + 2)$.

Conclusion

In this article, we have factored each of the given expressions using various factoring techniques. We have used the greatest common factor (GCF) method to factor out the common factors from each expression. The factored forms of the expressions are:

• $5x^2 + x = x(5x + 1)$
• $a^2 + 3a = a(a + 3)$
• $5n^2 + 2n = n(5n + 2)$
• $6n^2 + 3n = 3n(2n + 1)$
• $5n^2 - 10n = 5n(n - 2)$
• $3x^2 + 6x = 3x(x + 2)$
• $15x^2 + 30x = 15x(x + 2)$

In the previous article, we discussed how to factor each of the given expressions using various factoring techniques. However, we understand that you may still have some questions about factoring expressions. In this article, we will answer some of the most frequently asked questions about factoring expressions.

Q: What is factoring in mathematics?

A: Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It is an essential concept in algebra and is used to simplify complex expressions, solve equations, and find the roots of polynomials.

Q: What are the different types of factoring techniques?

A: There are several types of factoring techniques, including:

• Greatest Common Factor (GCF) method
• Difference of Squares method
• Sum and Difference of Cubes method
• Factoring by Grouping method
• Factoring Quadratic Expressions method

Q: How do I determine the greatest common factor (GCF) of two or more terms?

A: To determine the GCF of two or more terms, you need to find the largest expression that divides each term evenly. For example, the GCF of $5x^2$ and $x$ is $x$ because $x$ is the largest expression that divides both terms evenly.

Q: What is the difference of squares method?

A: The difference of squares method is a factoring technique used to factor expressions of the form $a^2 - b^2$. This method involves factoring the expression as $(a + b)(a - b)$.

Q: How do I factor expressions using the sum and difference of cubes method?

A: To factor expressions using the sum and difference of cubes method, you need to identify the expression as a sum or difference of cubes. For example, the expression $a^3 + b^3$ can be factored as $(a + b)(a^2 - ab + b^2)$.

Q: What is factoring by grouping?

A: Factoring by grouping is a factoring technique used to factor expressions that can be grouped into two or more parts. This method involves factoring each group separately and then combining the results.

Q: How do I factor quadratic expressions?

A: To factor quadratic expressions, you need to identify the expression as a product of two binomials. For example, the expression $x^2 + 5x + 6$ can be factored as $(x + 2)(x + 3)$.

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

A: Some common mistakes to avoid when factoring expressions include:

• Not identifying the greatest common factor (GCF) of two or more terms
• Not using the correct factoring technique for the given expression
• Not checking the factored form for errors
• Not simplifying the factored form

Q: How do I check the factored form for errors?

A: To check the factored form for errors, you need to multiply the factors together and compare the result with the original expression. If the result is the same as the original expression, then the factored form is correct.

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

A: Factoring expressions has many real-world applications, including:

• Simplifying complex expressions in physics and engineering
• Solving equations in finance and economics
• Finding the roots of polynomials in computer science and mathematics
• Modeling real-world phenomena in science and engineering

Conclusion

In this article, we have answered some of the most frequently asked questions about factoring expressions. We hope that this article has provided you with a better understanding of factoring expressions and how to use it to simplify complex expressions, solve equations, and find the roots of polynomials.