Simplify The Expression By Factoring Completely.${ 2r^3 - 54 = \square }$

Understanding the Problem

In this problem, we are given an algebraic expression in the form of an equation, where we need to simplify the expression by factoring completely. The given expression is $2r^3 - 54 = \square$. Our goal is to factorize the expression and simplify it to its simplest form.

What is Factoring?

Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down the expression into its prime factors, which are the basic building blocks of the expression. Factoring is an essential concept in algebra, and it plays a crucial role in solving equations and inequalities.

Step 1: Identify the Greatest Common Factor (GCF)

To factorize the expression, we need to identify the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides all the terms of the expression. In this case, the GCF of $2r^3$ and $54$ is $2$.

Step 2: Factor Out the GCF

Once we have identified the GCF, we can factor it out from the expression. To do this, we divide each term by the GCF and write the result as a product of the GCF and the remaining terms. In this case, we can factor out $2$ from both terms:

$$2r^3 - 54 = 2(r^3 - 27)$$

Step 3: Factor the Quadratic Expression

Now that we have factored out the GCF, we are left with a quadratic expression inside the parentheses. The quadratic expression is $r^3 - 27$. We can factor this expression by finding two numbers whose product is $-27$ and whose sum is $0$. These numbers are $-3$ and $9$, since $(-3) \times 9 = -27$ and $-3 + 9 = 0$.

Step 4: Write the Final Factored Form

Now that we have factored the quadratic expression, we can write the final factored form of the original expression. The final factored form is:

$$2r^3 - 54 = 2(r^3 - 27) = 2(r - 3)(r^2 + 3r + 9)$$

Conclusion

In this problem, we simplified the expression by factoring completely. We identified the GCF, factored it out, and then factored the quadratic expression inside the parentheses. The final factored form of the expression is $2(r - 3)(r^2 + 3r + 9)$. This form is simpler and more manageable than the original expression.

Tips and Tricks

• When factoring an expression, always look for the GCF first.
• When factoring a quadratic expression, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
• When writing the final factored form, make sure to include all the factors, including the GCF and the quadratic expression.

Common Mistakes to Avoid

• Not identifying the GCF correctly.
• Not factoring the quadratic expression correctly.
• Not including all the factors in the final factored form.

Real-World Applications

Factoring is an essential concept in algebra, and it has many real-world applications. Some examples include:

• Solving equations and inequalities.
• Graphing functions.
• Finding the roots of a polynomial.
• Solving systems of equations.

Practice Problems

Here are some practice problems to help you reinforce your understanding of factoring:

• Factor the expression $x^2 + 5x + 6$.
• Factor the expression $2x^2 - 7x - 3$.
• Factor the expression $x^3 - 8$.

Answer Key

• $x^2 + 5x + 6 = (x + 2)(x + 3)$
• $2x^2 - 7x - 3 = (2x + 1)(x - 3)$
• $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$
  Simplify the Expression by Factoring Completely: Q&A =====================================================

Q: What is factoring in algebra?

A: Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down the expression into its prime factors, which are the basic building blocks of the expression.

Q: Why is factoring important in algebra?

A: Factoring is an essential concept in algebra, and it plays a crucial role in solving equations and inequalities. It helps us to simplify complex expressions, identify the roots of a polynomial, and solve systems of equations.

Q: How do I identify the greatest common factor (GCF) of an expression?

A: To identify the GCF, look for the largest factor that divides all the terms of the expression. You can use the following steps:

1. List all the factors of each term.
2. Identify the common factors among the terms.
3. Choose the largest common factor as the GCF.

Q: How do I factor out the GCF from an expression?

A: To factor out the GCF, divide each term by the GCF and write the result as a product of the GCF and the remaining terms. For example, if the GCF is 2, you can factor out 2 from both terms as follows:

$$2r^3 - 54 = 2(r^3 - 27)$$

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, if the quadratic expression is $r^2 + 5r + 6$, you can factor it as follows:

$$r^2 + 5r + 6 = (r + 2)(r + 3)$$

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

A: Some common mistakes to avoid when factoring include:

• Not identifying the GCF correctly.
• Not factoring the quadratic expression correctly.
• Not including all the factors in the final factored form.

Q: How do I know if an expression can be factored?

A: An expression can be factored if it can be expressed as a product of simpler expressions. You can use the following steps to determine if an expression can be factored:

1. Look for the GCF of the terms.
2. Check if the quadratic expression can be factored.
3. If the expression cannot be factored, it may be a prime expression.

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

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

• Solving equations and inequalities.
• Graphing functions.
• Finding the roots of a polynomial.
• Solving systems of equations.

Q: How can I practice factoring?

A: You can practice factoring by working on the following exercises:

• Factor the expression $x^2 + 5x + 6$.
• Factor the expression $2x^2 - 7x - 3$.
• Factor the expression $x^3 - 8$.

Answer Key

• $x^2 + 5x + 6 = (x + 2)(x + 3)$
• $2x^2 - 7x - 3 = (2x + 1)(x - 3)$
• $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$

Conclusion

Factoring is an essential concept in algebra, and it plays a crucial role in solving equations and inequalities. By understanding how to factor expressions, you can simplify complex expressions, identify the roots of a polynomial, and solve systems of equations. Practice factoring by working on the exercises provided, and you will become proficient in this important algebraic concept.