Factor The Following Binomial.$\[ 25x^2 - 16 \\]$\[ ([?]x + \square)(\square X - \square) \\]

Introduction

Factoring binomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we will focus on factoring the binomial $25x^2 - 16$ using the method of factoring by grouping. This method involves grouping the terms of the quadratic expression in a way that allows us to factor out a common factor.

Understanding the Problem

The given binomial is $25x^2 - 16$. Our goal is to factor this expression into the form $([?]x + \square)(\square x - \square)$. To do this, we need to identify the two binomials that, when multiplied together, give us the original expression.

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factoring the binomial is to identify the greatest common factor (GCF) of the two terms. In this case, the GCF of $25x^2$ and $-16$ is $1$, since there is no common factor that divides both terms.

Step 2: Factor by Grouping

Since the GCF is $1$, we can proceed to factor the binomial by grouping. We can group the terms as follows:

$$25x^2 - 16 = (25x^2 + 0x + 0) - (0x + 0 + 16)$$

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

$$25x^2 - 16 = (5x)^2 - 4^2$$

Step 3: Apply the Difference of Squares Formula

The expression $(5x)^2 - 4^2$ is a difference of squares, which can be factored using the formula:

$$a^2 - b^2 = (a + b)(a - b)$$

In this case, $a = 5x$ and $b = 4$. Therefore, we can factor the expression as follows:

$$(5x)^2 - 4^2 = (5x + 4)(5x - 4)$$

Conclusion

In this article, we have factored the binomial $25x^2 - 16$ using the method of factoring by grouping. We identified the greatest common factor (GCF) of the two terms, grouped the terms, and applied the difference of squares formula to factor the expression. The final factored form of the binomial is $(5x + 4)(5x - 4)$.

Example Problems

Problem 1

Factor the binomial $9x^2 - 16$.

Solution

To factor the binomial $9x^2 - 16$, we can follow the same steps as before:

$$9x^2 - 16 = (9x^2 + 0x + 0) - (0x + 0 + 16)$$

$$9x^2 - 16 = (3x)^2 - 4^2$$

$$(3x)^2 - 4^2 = (3x + 4)(3x - 4)$$

Therefore, the factored form of the binomial $9x^2 - 16$ is $(3x + 4)(3x - 4)$.

Problem 2

Factor the binomial $x^2 - 49$.

Solution

To factor the binomial $x^2 - 49$, we can follow the same steps as before:

$$x^2 - 49 = (x^2 + 0x + 0) - (0x + 0 + 49)$$

$$x^2 - 49 = x^2 - 7^2$$

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

Therefore, the factored form of the binomial $x^2 - 49$ is $(x + 7)(x - 7)$.

Tips and Tricks

• When factoring a binomial, always look for the greatest common factor (GCF) of the two terms.
• Use the method of factoring by grouping to factor binomials that do not have a common factor.
• Apply the difference of squares formula to factor expressions of the form $a^2 - b^2$.

Conclusion

$$$$

Introduction

Factoring binomials 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 the binomial $25x^2 - 16$. In this article, we will answer some of the most frequently asked questions about factoring binomials.

Q&A

Q: What is the greatest common factor (GCF) of two terms?

A: The greatest common factor (GCF) of two terms is the largest factor that divides both terms without leaving a remainder.

Q: How do I factor a binomial that does not have a common factor?

A: To factor a binomial that does not have a common factor, you can use the method of factoring by grouping. This involves grouping the terms in a way that allows you to factor out a common factor.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

$$a^2 - b^2 = (a + b)(a - b)$$

This formula can be used to factor expressions of the form $a^2 - b^2$.

Q: How do I factor a binomial that is a difference of squares?

A: To factor a binomial that is a difference of squares, you can use the difference of squares formula. This involves identifying the values of $a$ and $b$ and then factoring the expression as $(a + b)(a - b)$.

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

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

• Not identifying the greatest common factor (GCF) of the two terms
• Not using the method of factoring by grouping when the binomial does not have a common factor
• Not applying the difference of squares formula when the binomial is a difference of squares
• Not checking the factored form to ensure that it is correct

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

A: To check the factored form of a binomial, you can multiply the two binomials together and see if you get the original expression. If you do, then the factored form is correct.

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

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

• Solving quadratic equations
• Finding the roots of a quadratic equation
• Graphing quadratic functions
• Solving systems of equations

Example Problems

Problem 1

Factor the binomial $9x^2 - 16$.

Solution

To factor the binomial $9x^2 - 16$, we can follow the same steps as before:

$$9x^2 - 16 = (9x^2 + 0x + 0) - (0x + 0 + 16)$$

$$9x^2 - 16 = (3x)^2 - 4^2$$

$$(3x)^2 - 4^2 = (3x + 4)(3x - 4)$$

Therefore, the factored form of the binomial $9x^2 - 16$ is $(3x + 4)(3x - 4)$.

Problem 2

Factor the binomial $x^2 - 49$.

Solution

To factor the binomial $x^2 - 49$, we can follow the same steps as before:

$$x^2 - 49 = (x^2 + 0x + 0) - (0x + 0 + 49)$$

$$x^2 - 49 = x^2 - 7^2$$

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

Therefore, the factored form of the binomial $x^2 - 49$ is $(x + 7)(x - 7)$.

Tips and Tricks

• When factoring a binomial, always look for the greatest common factor (GCF) of the two terms.
• Use the method of factoring by grouping to factor binomials that do not have a common factor.
• Apply the difference of squares formula to factor expressions of the form $a^2 - b^2$.
• Check the factored form to ensure that it is correct.

Conclusion

Factoring binomials is an essential skill in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we have answered some of the most frequently asked questions about factoring binomials. We have also provided example problems and tips and tricks to help you master the art of factoring binomials.