Which Of The Following Is The Correct Factorization Of The Trinomial Below? { -7x^2 - 5x + 18$}$A. { (-7x - 9)(x + 2)$}$B. { -1(7x - 9)(x + 2)$}$C. { -7(x - 6)(x + 1)$}$D. { (-7x + 9)(x - 2)$}$

Introduction

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we will explore the correct factorization of the trinomial $-7x^2 - 5x + 18$ and provide a step-by-step guide on how to factor trinomials.

Understanding Trinomials

A trinomial is a quadratic expression that consists of three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In the given trinomial $-7x^2 - 5x + 18$, the coefficients are $a = -7$, $b = -5$, and $c = 18$.

Factoring Trinomials

To factor a trinomial, we need to find two binomials whose product is equal to the trinomial. The general form of a binomial is $ax + b$. We can start by finding the factors of the constant term $c$ and the coefficient of the linear term $b$. In this case, the factors of $18$ are $1, 2, 3, 6, 9, 18$ and the factors of $-5$ are $1, -1, 5, -5$.

Step 1: Find the Factors of the Constant Term

The constant term is $18$. We need to find two numbers whose product is equal to $18$ and whose sum is equal to the coefficient of the linear term, which is $-5$. Let's try to find the factors of $18$ that add up to $-5$.

Factors of 18	Sum
1, 18	19
2, 9	11
3, 6	9

As we can see, the factors $-9$ and $2$ add up to $-5$. Therefore, we can write the trinomial as $-7x^2 - 9x + 2x + 18$.

Step 2: Factor by Grouping

Now that we have written the trinomial as $-7x^2 - 9x + 2x + 18$, we can factor by grouping. We can group the first two terms and the last two terms separately.

$-7x^2 - 9x + 2x + 18 = (-7x^2 - 9x) + (2x + 18)$

Now, we can factor out the common term from each group.

$(-7x^2 - 9x) + (2x + 18) = -7x(x + 3) + 2(x + 9)$

Step 3: Factor the Binomials

Now that we have factored the trinomial by grouping, we can factor the binomials separately.

$-7x(x + 3) + 2(x + 9) = -7x(x + 3) + 2(x + 9)$

We can see that the binomials $-7x(x + 3)$ and $2(x + 9)$ are already factored.

Conclusion

In conclusion, the correct factorization of the trinomial $-7x^2 - 5x + 18$ is $-7x(x + 3) + 2(x + 9)$. This can be further simplified to $(-7x - 9)(x + 2)$.

Answer

The correct answer is:

• A. [$(-7x - 9)(x + 2)$]

Discussion

Factoring trinomials is an important concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we have provided a step-by-step guide on how to factor trinomials. We have also discussed the importance of factoring trinomials and how it can be used to solve quadratic equations.

Common Mistakes

When factoring trinomials, there are several common mistakes that students make. Some of these mistakes include:

• Not factoring the trinomial correctly
• Not using the correct method to factor the trinomial
• Not checking the answer for accuracy

Tips and Tricks

Here are some tips and tricks that can help you factor trinomials correctly:

• Make sure to use the correct method to factor the trinomial
• Check the answer for accuracy
• Use the distributive property to check the answer
• Use the factoring method to check the answer

Conclusion

Introduction

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In our previous article, we provided a step-by-step guide on how to factor trinomials. In this article, we will answer some of the most frequently asked questions about factoring trinomials.

Q: What is a trinomial?

A trinomial is a quadratic expression that consists of three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I factor a trinomial?

To factor a trinomial, you need to find two binomials whose product is equal to the trinomial. The general form of a binomial is $ax + b$. You can start by finding the factors of the constant term $c$ and the coefficient of the linear term $b$.

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

When factoring trinomials, there are several common mistakes that students make. Some of these mistakes include:

• Not factoring the trinomial correctly
• Not using the correct method to factor the trinomial
• Not checking the answer for accuracy

Q: How do I check if my answer is correct?

To check if your answer is correct, you can use the distributive property to multiply the two binomials together and see if you get the original trinomial.

Q: What are some tips and tricks for factoring trinomials?

Here are some tips and tricks that can help you factor trinomials correctly:

• Make sure to use the correct method to factor the trinomial
• Check the answer for accuracy
• Use the distributive property to check the answer
• Use the factoring method to check the answer

Q: Can I factor a trinomial with a negative coefficient?

Yes, you can factor a trinomial with a negative coefficient. To do this, you need to factor the trinomial as if it were a positive coefficient, and then multiply the entire expression by -1.

Q: How do I factor a trinomial with a zero coefficient?

If the coefficient of the quadratic term is zero, then the trinomial is already factored. For example, the trinomial $0x^2 + 5x + 6$ is already factored as $(5x + 6)$.

Q: Can I factor a trinomial with a variable coefficient?

Yes, you can factor a trinomial with a variable coefficient. To do this, you need to factor the trinomial as if the variable coefficient were a constant, and then use the factoring method to factor the trinomial.

Conclusion

In conclusion, factoring trinomials is an important concept in algebra that involves expressing a quadratic expression as a product of two binomials. By following the tips and tricks provided in this article, you can improve your skills in factoring trinomials and become proficient in algebra.

Common Trinomial Factorization Mistakes

Here are some common mistakes to avoid when factoring trinomials:

• Not factoring the trinomial correctly
• Not using the correct method to factor the trinomial
• Not checking the answer for accuracy

Trinomial Factorization Examples

Here are some examples of trinomial factorization:

• $x^2 + 5x + 6 = (x + 3)(x + 2)$
• $x^2 - 7x + 12 = (x - 3)(x - 4)$
• $x^2 + 2x + 1 = (x + 1)^2$

Trinomial Factorization Practice

Here are some practice problems to help you improve your skills in factoring trinomials:

• Factor the trinomial $x^2 + 3x + 2$
• Factor the trinomial $x^2 - 4x + 4$
• Factor the trinomial $x^2 + 2x + 1$

Conclusion

In conclusion, factoring trinomials is an important concept in algebra that involves expressing a quadratic expression as a product of two binomials. By following the tips and tricks provided in this article, you can improve your skills in factoring trinomials and become proficient in algebra.