Factor The Expression: $25x^2 - 16$

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Introduction

In algebra, factoring is a process of expressing an algebraic expression as a product of simpler expressions. It is an essential concept in mathematics, and it plays a crucial role in solving equations and inequalities. In this article, we will focus on factoring the expression $25x^2 - 16$. We will explore the different methods of factoring and provide step-by-step solutions to help you understand the concept better.

What is Factoring?

Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It involves finding the factors of an expression, which are the numbers or variables that multiply together to give the original expression. Factoring is an essential concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics.

Methods of Factoring

There are several methods of factoring, including:

• Factoring out the greatest common factor (GCF): This method involves factoring out the greatest common factor of the terms in the expression.
• Factoring by grouping: This method involves grouping the terms in the expression into pairs and factoring out the common factors from each pair.
• Factoring quadratic expressions: This method involves factoring quadratic expressions of the form $ax^2 + bx + c$.

Factoring the Expression $25x^2 - 16$

To factor the expression $25x^2 - 16$, we can use the method of factoring out the greatest common factor (GCF). The GCF of the terms $25x^2$ and $-16$ is $1$, but we can factor out a $1$ from each term.

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

Now, we can use the difference of squares formula to factor the expression:

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

Therefore, the factored form of the expression $25x^2 - 16$ is $(5x + 4)(5x - 4)$.

Example Problems

Here are some example problems to help you practice factoring:

• Factor the expression $9x^2 - 16$.
• Factor the expression $x^2 + 5x + 6$.
• Factor the expression $x^2 - 7x + 12$.

Solutions

• To factor the expression $9x^2 - 16$, we can use the method of factoring out the greatest common factor (GCF). The GCF of the terms $9x^2$ and $-16$ is $1$, but we can factor out a $1$ from each term.

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

Now, we can use the difference of squares formula to factor the expression:

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

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

• To factor the expression $x^2 + 5x + 6$, we can use the method of factoring by grouping. We can group the terms $x^2$ and $5x$ together, and the term $6$ by itself.

$x^2 + 5x + 6 = (x^2 + 5x) + 6$

Now, we can factor out the common factor $x$ from the first two terms:

$(x^2 + 5x) + 6 = x(x + 5) + 6$

Next, we can factor out the common factor $1$ from the last two terms:

$x(x + 5) + 6 = x(x + 5) + 2 \cdot 3$

Now, we can factor out the common factor $2$ from the last two terms:

$x(x + 5) + 2 \cdot 3 = x(x + 5) + 2(3)$

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

• To factor the expression $x^2 - 7x + 12$, we can use the method of factoring by grouping. We can group the terms $x^2$ and $-7x$ together, and the term $12$ by itself.

$x^2 - 7x + 12 = (x^2 - 7x) + 12$

Now, we can factor out the common factor $x$ from the first two terms:

$(x^2 - 7x) + 12 = x(x - 7) + 12$

Next, we can factor out the common factor $1$ from the last two terms:

$x(x - 7) + 12 = x(x - 7) + 3 \cdot 4$

Now, we can factor out the common factor $3$ from the last two terms:

$x(x - 7) + 3 \cdot 4 = x(x - 7) + 3(4)$

Therefore, the factored form of the expression $x^2 - 7x + 12$ is $x(x - 7) + 3(4)$.

Conclusion

In conclusion, factoring is an essential concept in mathematics, and it plays a crucial role in solving equations and inequalities. In this article, we have explored the different methods of factoring, including factoring out the greatest common factor (GCF), factoring by grouping, and factoring quadratic expressions. We have also provided step-by-step solutions to help you understand the concept better. With practice and patience, you can master the art of factoring and become proficient in solving algebraic expressions.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Glossary

• Factoring: The process of expressing an algebraic expression as a product of simpler expressions.
• Greatest common factor (GCF): The largest factor that divides all the terms in an expression.
• Difference of squares formula: A formula that states that $a^2 - b^2 = (a + b)(a - b)$.
• Factoring by grouping: A method of factoring that involves grouping the terms in an expression into pairs and factoring out the common factors from each pair.
  Q&A: Factoring Expressions =============================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about factoring expressions. Whether you are a student, a teacher, or a professional, you will find the answers to your questions here.

Q: What is factoring?

A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions. It involves finding the factors of an expression, which are the numbers or variables that multiply together to give the original expression.

Q: Why is factoring important?

A: Factoring is an essential concept in mathematics, and it plays a crucial role in solving equations and inequalities. It helps us to simplify complex expressions, identify patterns, and solve problems more efficiently.

Q: What are the different methods of factoring?

A: There are several methods of factoring, including:

• Factoring out the greatest common factor (GCF): This method involves factoring out the greatest common factor of the terms in the expression.
• Factoring by grouping: This method involves grouping the terms in the expression into pairs and factoring out the common factors from each pair.
• Factoring quadratic expressions: This method involves factoring quadratic expressions of the form $ax^2 + bx + c$.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you can use the method of factoring out the greatest common factor (GCF), factoring by grouping, or factoring using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that states that for an expression of the form $ax^2 + bx + c$, the solutions are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I factor a difference of squares?

A: To factor a difference of squares, you can use the formula:

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

Q: What is the difference of squares formula?

A: The difference of squares formula is a formula that states that for an expression of the form $a^2 - b^2$, the expression can be factored as:

$$(a + b)(a - b)$$

Q: How do I factor a sum of squares?

A: To factor a sum of squares, you cannot factor it in the same way as a difference of squares. However, you can use the formula:

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

where $i$ is the imaginary unit.

Q: What is the imaginary unit?

A: The imaginary unit is a complex number that is defined as:

$$i = \sqrt{-1}$$

Q: How do I factor a rational expression?

A: To factor a rational expression, you can use the method of factoring out the greatest common factor (GCF) or factoring by grouping.

Q: What is a rational expression?

A: A rational expression is an expression that can be written in the form:

$$\frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomials.

Q: How do I factor a polynomial?

A: To factor a polynomial, you can use the method of factoring out the greatest common factor (GCF), factoring by grouping, or factoring using the quadratic formula.

Conclusion

In conclusion, factoring is an essential concept in mathematics, and it plays a crucial role in solving equations and inequalities. We have answered some of the most frequently asked questions about factoring expressions, and we hope that this article has been helpful to you.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Glossary

• Factoring: The process of expressing an algebraic expression as a product of simpler expressions.
• Greatest common factor (GCF): The largest factor that divides all the terms in an expression.
• Difference of squares formula: A formula that states that $a^2 - b^2 = (a + b)(a - b)$.
• Factoring by grouping: A method of factoring that involves grouping the terms in an expression into pairs and factoring out the common factors from each pair.
• Quadratic formula: A formula that states that for an expression of the form $ax^2 + bx + c$, the solutions are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Imaginary unit: A complex number that is defined as:

$$i = \sqrt{-1}$$

• Rational expression: An expression that can be written in the form:

$$\frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomials.