Factor The Trinomial $x^2 + X - 42$.1. Use The Completed X-diagram To Replace The $x$-term In The Trinomial With Two $x$-terms: $\[x^2 + X - 42 = X^2 + (b)x + (c)x - 42\\]2. Next, Use Double Grouping To Factor The

Introduction

Factoring trinomials is a fundamental concept in algebra that can be used to simplify complex expressions and solve equations. In this article, we will focus on factoring the trinomial $x^2 + x - 42$ using the completed X-diagram and double grouping methods.

Method 1: Completed X-Diagram

The completed X-diagram is a visual tool used to factor trinomials. It involves replacing the $x$-term in the trinomial with two $x$-terms, which are then factored using the diagram.

Step 1: Replace the $x$-term with two $x$-terms

To factor the trinomial $x^2 + x - 42$, we start by replacing the $x$-term with two $x$-terms:

$$x^2 + x - 42 = x^2 + (b)x + (c)x - 42$$

Step 2: Identify the values of $b$ and $c$

We need to find the values of $b$ and $c$ such that the two $x$-terms multiply to give the constant term $-42$. We can do this by listing the factors of $-42$ and finding the pairs that add up to $1$.

The factors of $-42$ are: $-1, 1, -2, 2, -3, 3, -6, 6, -7, 7, -14, 14, -21, 21, -42, 42$

We can see that the pairs $(-1, -42)$ and $(1, -42)$ add up to $-1$ and $-1$ respectively. Therefore, we can set $b = -1$ and $c = -42$.

Step 3: Factor the trinomial

Now that we have the values of $b$ and $c$, we can factor the trinomial:

$$x^2 + x - 42 = x^2 - x - 42$$

We can factor the trinomial by grouping the terms:

$$(x^2 - x) - 42$$

We can factor the first two terms by taking out a common factor of $x$:

$$x(x - 1) - 42$$

We can factor the last two terms by taking out a common factor of $-1$:

$$x(x - 1) - 42 = (x - 7)(x + 6)$$

Therefore, the factored form of the trinomial $x^2 + x - 42$ is $(x - 7)(x + 6)$.

Method 2: Double Grouping

Double grouping is another method used to factor trinomials. It involves grouping the terms in pairs and factoring each pair separately.

Step 1: Group the terms in pairs

To factor the trinomial $x^2 + x - 42$, we start by grouping the terms in pairs:

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

Step 2: Factor each pair separately

We can factor the first pair by taking out a common factor of $x$:

$$x(x + 1) - 42$$

We can factor the last two terms by taking out a common factor of $-1$:

$$x(x + 1) - 42 = (x - 7)(x + 6)$$

Therefore, the factored form of the trinomial $x^2 + x - 42$ is $(x - 7)(x + 6)$.

Conclusion

In this article, we have discussed two methods for factoring trinomials: the completed X-diagram and double grouping. We have used these methods to factor the trinomial $x^2 + x - 42$ and obtained the factored form $(x - 7)(x + 6)$. These methods can be used to factor a wide range of trinomials and are an essential part of algebra.

Example Problems

Problem 1

Factor the trinomial $x^2 + 5x - 6$.

Solution

We can use the completed X-diagram to factor the trinomial:

$$x^2 + 5x - 6 = x^2 + (b)x + (c)x - 6$$

We can identify the values of $b$ and $c$ by listing the factors of $-6$ and finding the pairs that add up to $5$.

The factors of $-6$ are: $-1, 1, -2, 2, -3, 3, -6, 6$

We can see that the pairs $(-1, -6)$ and $(1, -6)$ add up to $-1$ and $-1$ respectively. Therefore, we can set $b = -1$ and $c = -6$.

We can factor the trinomial by grouping the terms:

$$(x^2 - x) - 6$$

We can factor the first two terms by taking out a common factor of $x$:

$$x(x - 1) - 6$$

We can factor the last two terms by taking out a common factor of $-1$:

$$x(x - 1) - 6 = (x - 3)(x + 2)$$

Therefore, the factored form of the trinomial $x^2 + 5x - 6$ is $(x - 3)(x + 2)$.

Problem 2

Factor the trinomial $x^2 - 3x - 18$.

Solution

We can use the double grouping method to factor the trinomial:

$$(x^2 - 3x) - 18$$

We can factor the first two terms by taking out a common factor of $x$:

$$x(x - 3) - 18$$

We can factor the last two terms by taking out a common factor of $-1$:

$$x(x - 3) - 18 = (x - 6)(x + 3)$$

Therefore, the factored form of the trinomial $x^2 - 3x - 18$ is $(x - 6)(x + 3)$.

Glossary

• Trinomial: A polynomial with three terms.
• Factoring: The process of expressing a polynomial as a product of simpler polynomials.
• Completed X-diagram: A visual tool used to factor trinomials.
• Double grouping: A method used to factor trinomials by grouping the terms in pairs and factoring each pair separately.
  Factoring Trinomials: A Q&A Guide =====================================

Introduction

Factoring trinomials is a fundamental concept in algebra that can be used to simplify complex expressions and solve equations. 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 polynomial with three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

Q: What is factoring?

Factoring is the process of expressing a polynomial as a product of simpler polynomials. In other words, it is the process of breaking down a polynomial into its constituent parts.

Q: What are the different methods of factoring trinomials?

There are several methods of factoring trinomials, including:

• Completed X-diagram: A visual tool used to factor trinomials.
• Double grouping: A method used to factor trinomials by grouping the terms in pairs and factoring each pair separately.
• Grouping: A method used to factor trinomials by grouping the terms in pairs and factoring each pair separately.
• Factoring by grouping with a common factor: A method used to factor trinomials by grouping the terms in pairs and factoring each pair separately, with a common factor.

Q: How do I use the completed X-diagram to factor a trinomial?

To use the completed X-diagram to factor a trinomial, follow these steps:

1. Write the trinomial in the form $ax^2 + bx + c$.
2. Draw a diagram with two rows and two columns.
3. In the top row, write the terms of the trinomial.
4. In the bottom row, write the factors of the constant term $c$.
5. Identify the values of $b$ and $c$ by listing the factors of $c$ and finding the pairs that add up to $b$.
6. Factor the trinomial by grouping the terms.

Q: How do I use double grouping to factor a trinomial?

To use double grouping to factor a trinomial, follow these steps:

1. Write the trinomial in the form $ax^2 + bx + c$.
2. Group the terms in pairs.
3. Factor each pair separately.
4. Factor the last two terms by taking out a common factor.

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

Some common mistakes to avoid when factoring trinomials include:

• Not identifying the values of $b$ and $c$ correctly: Make sure to list the factors of $c$ and find the pairs that add up to $b$.
• Not factoring the trinomial correctly: Make sure to factor the trinomial by grouping the terms and factoring each pair separately.
• Not checking the factored form: Make sure to check the factored form by multiplying the factors together.

Q: How do I check the factored form of a trinomial?

To check the factored form of a trinomial, follow these steps:

1. Multiply the factors together.
2. Simplify the expression.
3. Check that the result is equal to the original trinomial.

Conclusion

In this article, we have answered some of the most frequently asked questions about factoring trinomials. We have discussed the different methods of factoring trinomials, including the completed X-diagram and double grouping. We have also provided some tips and tricks for factoring trinomials and checking the factored form.

Example Problems

Problem 1

Factor the trinomial $x^2 + 5x - 6$.

Solution

We can use the completed X-diagram to factor the trinomial:

$$x^2 + 5x - 6 = x^2 + (b)x + (c)x - 6$$

We can identify the values of $b$ and $c$ by listing the factors of $-6$ and finding the pairs that add up to $5$.

The factors of $-6$ are: $-1, 1, -2, 2, -3, 3, -6, 6$

We can see that the pairs $(-1, -6)$ and $(1, -6)$ add up to $-1$ and $-1$ respectively. Therefore, we can set $b = -1$ and $c = -6$.

We can factor the trinomial by grouping the terms:

$$(x^2 - x) - 6$$

We can factor the first two terms by taking out a common factor of $x$:

$$x(x - 1) - 6$$

We can factor the last two terms by taking out a common factor of $-1$:

$$x(x - 1) - 6 = (x - 3)(x + 2)$$

Therefore, the factored form of the trinomial $x^2 + 5x - 6$ is $(x - 3)(x + 2)$.

Problem 2

Factor the trinomial $x^2 - 3x - 18$.

Solution

We can use the double grouping method to factor the trinomial:

$$(x^2 - 3x) - 18$$

We can factor the first two terms by taking out a common factor of $x$:

$$x(x - 3) - 18$$

We can factor the last two terms by taking out a common factor of $-1$:

$$x(x - 3) - 18 = (x - 6)(x + 3)$$

Therefore, the factored form of the trinomial $x^2 - 3x - 18$ is $(x - 6)(x + 3)$.

Glossary

• Trinomial: A polynomial with three terms.
• Factoring: The process of expressing a polynomial as a product of simpler polynomials.
• Completed X-diagram: A visual tool used to factor trinomials.
• Double grouping: A method used to factor trinomials by grouping the terms in pairs and factoring each pair separately.