Factor The Trinomial.${ C^3 - 2c^2 - 8c }$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. { C^3 - 2c^2 - 8c = $}$ { \Box$}$ (Factor Completely.)B. The Trinomial Is Not

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Understanding the Problem

Factoring a trinomial is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring the given trinomial: c3−2c2−8cc^3 - 2c^2 - 8c. Our goal is to identify the correct choice and, if necessary, fill in the answer box to complete our choice.

What is Factoring?

Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It involves finding the factors of a polynomial that, when multiplied together, result in the original polynomial. Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and identify the roots of a polynomial.

The Given Trinomial

The given trinomial is c3−2c2−8cc^3 - 2c^2 - 8c. To factor this trinomial, we need to identify the greatest common factor (GCF) of the three terms. The GCF is the largest expression that divides each term without leaving a remainder.

Step 1: Identify the GCF

The GCF of the three terms is cc. We can factor out cc from each term:

c3−2c2−8c=c(c2−2c−8)c^3 - 2c^2 - 8c = c(c^2 - 2c - 8)

Step 2: Factor the Quadratic Expression

The quadratic expression c2−2c−8c^2 - 2c - 8 can be factored using the factoring method. We need to find two numbers whose product is −8-8 and whose sum is −2-2. These numbers are −4-4 and 22. Therefore, we can write:

c2−2c−8=(c−4)(c+2)c^2 - 2c - 8 = (c - 4)(c + 2)

Step 3: Write the Factored Form

Now that we have factored the quadratic expression, we can write the factored form of the original trinomial:

c3−2c2−8c=c(c−4)(c+2)c^3 - 2c^2 - 8c = c(c - 4)(c + 2)

Conclusion

In conclusion, the factored form of the given trinomial is c(c−4)(c+2)c(c - 4)(c + 2). This is the correct choice, and we can fill in the answer box as follows:

A. c3−2c2−8c=c(c−4)(c+2)c^3 - 2c^2 - 8c = c(c - 4)(c + 2)

Discussion

Factoring a trinomial involves identifying the GCF and factoring the quadratic expression. In this case, we factored out cc from each term and then factored the quadratic expression using the factoring method. The factored form of the trinomial is c(c−4)(c+2)c(c - 4)(c + 2).

Common Mistakes

When factoring a trinomial, it's essential to identify the GCF and factor the quadratic expression correctly. Some common mistakes include:

  • Failing to identify the GCF
  • Factoring the quadratic expression incorrectly
  • Not checking the factored form for correctness

Tips and Tricks

To factor a trinomial correctly, follow these tips and tricks:

  • Identify the GCF first
  • Factor the quadratic expression using the factoring method
  • Check the factored form for correctness
  • Use the factored form to simplify complex expressions and solve equations

Conclusion

Q: What is the greatest common factor (GCF) of the three terms in the trinomial c3−2c2−8cc^3 - 2c^2 - 8c?

A: The GCF of the three terms is cc. We can factor out cc from each term to get c(c2−2c−8)c(c^2 - 2c - 8).

Q: How do I factor the quadratic expression c2−2c−8c^2 - 2c - 8?

A: To factor the quadratic expression, we need to find two numbers whose product is −8-8 and whose sum is −2-2. These numbers are −4-4 and 22. Therefore, we can write:

c2−2c−8=(c−4)(c+2)c^2 - 2c - 8 = (c - 4)(c + 2)

Q: What is the factored form of the trinomial c3−2c2−8cc^3 - 2c^2 - 8c?

A: The factored form of the trinomial is c(c−4)(c+2)c(c - 4)(c + 2).

Q: How do I check if the factored form is correct?

A: To check if the factored form is correct, we need to multiply the factors together and see if we get the original trinomial. In this case, we can multiply c(c−4)(c+2)c(c - 4)(c + 2) together to get:

c(c−4)(c+2)=c(c2−2c−8)=c3−2c2−8cc(c - 4)(c + 2) = c(c^2 - 2c - 8) = c^3 - 2c^2 - 8c

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

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

  • Failing to identify the GCF
  • Factoring the quadratic expression incorrectly
  • Not checking the factored form for correctness

Q: How do I simplify complex expressions using the factored form?

A: To simplify complex expressions using the factored form, we can use the factored form to cancel out common factors. For example, if we have the expression:

c3−2c2−8cc\frac{c^3 - 2c^2 - 8c}{c}

We can simplify it by canceling out the common factor cc:

c3−2c2−8cc=c2−2c−8\frac{c^3 - 2c^2 - 8c}{c} = c^2 - 2c - 8

Q: How do I use the factored form to solve equations?

A: To use the factored form to solve equations, we can set the factored form equal to zero and solve for the variable. For example, if we have the equation:

c3−2c2−8c=0c^3 - 2c^2 - 8c = 0

We can set the factored form equal to zero and solve for cc:

c(c−4)(c+2)=0c(c - 4)(c + 2) = 0

This gives us three possible solutions: c=0c = 0, c=4c = 4, and c=−2c = -2.

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

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

  • Simplifying complex expressions in physics and engineering
  • Solving equations in economics and finance
  • Modeling population growth and decay in biology and ecology
  • Analyzing data in statistics and data science

Conclusion

Factoring trinomials is a fundamental concept in algebra that has many real-world applications. By understanding how to factor trinomials, we can simplify complex expressions, solve equations, and analyze data. In this article, we provided a step-by-step guide to factoring trinomials, as well as a Q&A section to help you understand the concept better.