Factor The Trinomial Completely By Using Any Method. Remember To Look For A Common Factor First. Select Prime If The Polynomial Cannot Be Factored. 5 U 2 + 31 U + 6 5u^2 + 31u + 6 5 U 2 + 31 U + 6 { \square$}$ Prime

Introduction

Factoring a trinomial is a fundamental concept in algebra that involves expressing a quadratic expression as a product of simpler expressions. In this article, we will focus on factoring the trinomial $5u^2 + 31u + 6$ completely using various methods. We will also discuss the importance of looking for a common factor first and provide a step-by-step guide on how to factor the given trinomial.

Understanding the Trinomial

A trinomial is a quadratic expression that consists of three terms. In the given trinomial $5u^2 + 31u + 6$, the first term is $5u^2$, the second term is $31u$, and the third term is $6$. To factor this trinomial, we need to find two binomials whose product equals the given trinomial.

Looking for a Common Factor

Before we start factoring the trinomial, we need to look for a common factor. A common factor is a factor that divides all the terms of the trinomial. In this case, we can see that the greatest common factor (GCF) of the three terms is $1$. However, we can also look for other common factors such as $u$ or $5$. Since $u$ is a common factor of the first two terms, we can factor it out:

$$5u^2 + 31u + 6 = u(5u + 31) + 6$$

However, we cannot factor out $5$ from the first two terms. Therefore, we will move on to the next step.

Factoring by Grouping

One method of factoring a trinomial is by grouping. This involves grouping the first two terms and the last two terms together and then factoring out a common factor from each group. In this case, we can group the first two terms and the last two terms as follows:

$$5u^2 + 31u + 6 = (5u^2 + 31u) + 6$$

Now, we can factor out a common factor from each group:

$$(5u^2 + 31u) + 6 = 5u(u + 6) + 6$$

However, we cannot factor out a common factor from the last two terms. Therefore, we will move on to the next step.

Factoring by Using the AC Method

Another method of factoring a trinomial is by using the AC method. This involves finding two numbers whose product equals the product of the first and last terms and whose sum equals the middle term. In this case, we need to find two numbers whose product equals $5 \times 6 = 30$ and whose sum equals $31$. The two numbers are $10$ and $21$. Therefore, we can write the trinomial as follows:

$$5u^2 + 31u + 6 = 5u^2 + 10u + 21u + 6$$

Now, we can factor out a common factor from each group:

$$5u^2 + 10u + 21u + 6 = 5u(u + 2) + 21(u + 2)$$

However, we cannot factor out a common factor from the last two terms. Therefore, we will move on to the next step.

Factoring by Using the FOIL Method

The FOIL method is a method of factoring a trinomial by multiplying the first terms, the outer terms, the inner terms, and the last terms together. In this case, we can use the FOIL method as follows:

$$(5u + ?)(u + ?) = 5u^2 + 31u + 6$$

Multiplying the first terms, we get:

$$(5u)(u) = 5u^2$$

Multiplying the outer terms, we get:

$$(5u)(?) = 31u$$

Multiplying the inner terms, we get:

$$(?)(u) = 6$$

Multiplying the last terms, we get:

$$(?)(?) = 0$$

Now, we can solve for the unknown terms:

$$(5u)(?) = 31u \Rightarrow ? = \frac{31u}{5u} = \frac{31}{5}$$

$$(?)(u) = 6 \Rightarrow ? = \frac{6}{u}$$

Therefore, we can write the trinomial as follows:

$$(5u + \frac{31}{5})(u + \frac{6}{u}) = 5u^2 + 31u + 6$$

However, this is not a valid factorization since the second term is not a polynomial.

Conclusion

In this article, we have discussed the importance of looking for a common factor first when factoring a trinomial. We have also provided a step-by-step guide on how to factor the trinomial $5u^2 + 31u + 6$ using various methods. Unfortunately, we were unable to factor the trinomial completely using any of the methods. Therefore, we conclude that the trinomial cannot be factored and is therefore prime.

Final Answer

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Q: What is a trinomial?

A: A trinomial is a quadratic expression that consists of three terms. It is a polynomial of degree two, which means it has two variables raised to the power of two.

Q: Why is it important to look for a common factor first when factoring a trinomial?

A: Looking for a common factor first is important because it can simplify the factoring process and make it easier to factor the trinomial. If there is a common factor, it can be factored out of all the terms, making it easier to factor the remaining terms.

Q: What are some common methods for factoring trinomials?

A: There are several common methods for factoring trinomials, including:

• Factoring by grouping
• Factoring by using the AC method
• Factoring by using the FOIL method
• Factoring by using the difference of squares method

Q: What is the AC method?

A: The AC method is a method of factoring trinomials by finding two numbers whose product equals the product of the first and last terms and whose sum equals the middle term.

Q: What is the FOIL method?

A: The FOIL method is a method of factoring trinomials by multiplying the first terms, the outer terms, the inner terms, and the last terms together.

Q: Why is it difficult to factor some trinomials?

A: It can be difficult to factor some trinomials because they may not have a common factor, or they may not be able to be factored using the common methods. In some cases, the trinomial may be prime, meaning it cannot be factored further.

Q: What is a prime trinomial?

A: A prime trinomial is a trinomial that cannot be factored further. It is a trinomial that has no common factors and cannot be expressed as a product of simpler expressions.

Q: How can I determine if a trinomial is prime?

A: To determine if a trinomial is prime, you can try factoring it using the common methods. If you are unable to factor it, then it is likely prime.

Q: What are some tips for factoring trinomials?

A: Here are some tips for factoring trinomials:

• Look for a common factor first
• Use the AC method or the FOIL method
• Check for a difference of squares
• Check for a sum or difference of cubes
• Use a calculator or computer program to check your work

Q: Can I use a calculator or computer program to factor trinomials?

A: Yes, you can use a calculator or computer program to factor trinomials. Many calculators and computer programs have built-in functions for factoring polynomials.

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

A: Here are some common mistakes to avoid when factoring trinomials:

• Not looking for a common factor first
• Not using the correct method for factoring
• Not checking for a difference of squares or a sum or difference of cubes
• Not using a calculator or computer program to check your work

Q: How can I practice factoring trinomials?

A: You can practice factoring trinomials by working through examples and exercises in a textbook or online resource. You can also use a calculator or computer program to generate random trinomials for you to factor.