What Is The Factorization Of The Trinomial Below?$\[ -x^2 + X + 42 \\]A. \[$(-x + 6)(x + 7)\$\]B. \[$-1(x - 7)(x + 6)\$\]C. \[$(x + 6)(x - 7)\$\]D. \[$-1(x + 7)(x + 6)\$\]

Introduction

In algebra, factorization is a process of expressing a polynomial as a product of simpler polynomials. It is an essential concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will focus on the factorization of a trinomial, which is a polynomial with three terms. We will explore the different methods of factorization and apply them to a specific trinomial to find its factorization.

What is a Trinomial?

A trinomial is a polynomial with three terms. It can be written in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Trinomials can be factored using various methods, including the factoring by grouping method, the factoring by difference of squares method, and the factoring by sum and difference method.

Factoring by Grouping Method

The factoring by grouping method is a technique used to factor a trinomial by grouping its terms into two pairs. This method is useful when the trinomial has two terms with a common factor. To factor a trinomial using this method, we need to identify the two terms with a common factor and group them together. We then factor out the common factor from each group.

Factoring by Difference of Squares Method

The factoring by difference of squares method is a technique used to factor a trinomial that can be written in the form of $a^2 - b^2$. This method is useful when the trinomial has two terms that are perfect squares. To factor a trinomial using this method, we need to identify the two terms that are perfect squares and write them in the form of $a^2 - b^2$. We then factor the trinomial using the formula $(a + b)(a - b)$.

Factoring by Sum and Difference Method

The factoring by sum and difference method is a technique used to factor a trinomial that can be written in the form of $a^2 + 2ab + b^2$. This method is useful when the trinomial has two terms that are perfect squares and a common factor. To factor a trinomial using this method, we need to identify the two terms that are perfect squares and the common factor. We then factor the trinomial using the formula $(a + b)^2$.

Applying the Methods to the Trinomial

Now, let's apply the methods to the trinomial $-x^2 + x + 42$. We can see that the trinomial has three terms, and the first term is a perfect square. We can write the trinomial in the form of $a^2 - b^2$, where $a = -x$ and $b = 7$. We can then factor the trinomial using the formula $(a + b)(a - b)$.

Step 1: Identify the Two Terms that are Perfect Squares

The first term, $-x^2$, is a perfect square, and the second term, $x$, is not a perfect square. However, we can rewrite the second term as $x = 7 - 6$, where $7$ and $6$ are perfect squares.

Step 2: Write the Trinomial in the Form of $a^2 - b^2$

We can write the trinomial in the form of $a^2 - b^2$, where $a = -x$ and $b = 7$. We have:

$$-x^2 + x + 42 = (-x)^2 - 7^2$$

Step 3: Factor the Trinomial Using the Formula $(a + b)(a - b)$

We can now factor the trinomial using the formula $(a + b)(a - b)$. We have:

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

Step 4: Simplify the Expression

We can simplify the expression by multiplying the two factors together. We have:

$$(-x + 7)(-x - 7) = x^2 - 49$$

Conclusion

In this article, we have applied the factoring by difference of squares method to the trinomial $-x^2 + x + 42$. We have identified the two terms that are perfect squares, written the trinomial in the form of $a^2 - b^2$, and factored the trinomial using the formula $(a + b)(a - b)$. We have also simplified the expression to get the final answer.

Final Answer

The final answer is $\boxed{(-x + 7)(-x - 7)}$.

Introduction

In the previous article, we explored the factorization of the trinomial $-x^2 + x + 42$. We applied the factoring by difference of squares method to factor the trinomial and obtained the final answer $\boxed{(-x + 7)(-x - 7)}$. In this article, we will answer some frequently asked questions related to the factorization of the trinomial.

Q1: What is the factorization of the trinomial $-x^2 + x + 42$?

A1: The factorization of the trinomial $-x^2 + x + 42$ is $\boxed{(-x + 7)(-x - 7)}$.

Q2: How do I factor a trinomial using the factoring by difference of squares method?

A2: To factor a trinomial using the factoring by difference of squares method, you need to identify the two terms that are perfect squares and write them in the form of $a^2 - b^2$. You can then factor the trinomial using the formula $(a + b)(a - b)$.

Q3: What are the two terms that are perfect squares in the trinomial $-x^2 + x + 42$?

A3: The two terms that are perfect squares in the trinomial $-x^2 + x + 42$ are $-x^2$ and $7^2$.

Q4: How do I simplify the expression $(-x + 7)(-x - 7)$?

A4: To simplify the expression $(-x + 7)(-x - 7)$, you need to multiply the two factors together. You have:

$$(-x + 7)(-x - 7) = x^2 - 49$$

Q5: What is the final answer to the trinomial $-x^2 + x + 42$?

A5: The final answer to the trinomial $-x^2 + x + 42$ is $\boxed{(-x + 7)(-x - 7)}$.

Q6: Can I factor a trinomial using other methods?

A6: Yes, you can factor a trinomial using other methods, such as the factoring by grouping method, the factoring by sum and difference method, and the factoring by greatest common factor method.

Q7: How do I determine which method to use to factor a trinomial?

A7: To determine which method to use to factor a trinomial, you need to examine the trinomial and identify its characteristics. You can then choose the method that best suits the trinomial.

Q8: What are some common mistakes to avoid when factoring a trinomial?

A8: Some common mistakes to avoid when factoring a trinomial include:

• Not identifying the two terms that are perfect squares
• Not writing the trinomial in the correct form
• Not factoring the trinomial correctly
• Not simplifying the expression correctly

Conclusion

In this article, we have answered some frequently asked questions related to the factorization of the trinomial $-x^2 + x + 42$. We have provided the final answer, explained the factoring by difference of squares method, and identified some common mistakes to avoid when factoring a trinomial.

Final Answer

The final answer is $\boxed{(-x + 7)(-x - 7)}$.