Factor The Trinomial Completely:$\[ 10y^3 - 25y^2 - 125y \\]Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Answer:A. $\[ 10y^3 - 25y^2 - 125y = \, \square \\] (Factor Completely.)B. The

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Introduction

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 trinomial 10y3−25y2−125y10y^3 - 25y^2 - 125y completely. We will use a step-by-step approach to break down the problem and provide a clear understanding of the factoring process.

Understanding the Trinomial

A trinomial is a polynomial with three terms. In this case, the trinomial is 10y3−25y2−125y10y^3 - 25y^2 - 125y. To factor this trinomial, we need to identify the greatest common factor (GCF) of the three terms.

Identifying the Greatest Common Factor (GCF)

The GCF of the three terms is the largest factor that divides each term evenly. In this case, the GCF is 5y5y. We can factor out the GCF from each term to get:

10y3−25y2−125y=5y(2y2−5y−25)10y^3 - 25y^2 - 125y = 5y(2y^2 - 5y - 25)

Factoring the Quadratic Expression

The quadratic expression 2y2−5y−252y^2 - 5y - 25 can be factored using the quadratic formula or by finding two numbers whose product is −50-50 and whose sum is −5-5. The two numbers are −10-10 and 55, so we can write:

2y2−5y−25=(2y+5)(y−5)2y^2 - 5y - 25 = (2y + 5)(y - 5)

Factoring the Trinomial Completely

Now that we have factored the quadratic expression, we can factor the trinomial completely by multiplying the GCF with the factored quadratic expression:

10y3−25y2−125y=5y(2y+5)(y−5)10y^3 - 25y^2 - 125y = 5y(2y + 5)(y - 5)

Conclusion

Factoring a trinomial completely involves identifying the GCF and factoring the quadratic expression. By following the step-by-step approach outlined in this article, we can factor the trinomial 10y3−25y2−125y10y^3 - 25y^2 - 125y completely. The final factored form of the trinomial is 5y(2y+5)(y−5)5y(2y + 5)(y - 5).

Common Mistakes to Avoid

When factoring a trinomial, it's essential to avoid common mistakes such as:

  • Not identifying the GCF correctly
  • Not factoring the quadratic expression correctly
  • Not multiplying the GCF with the factored quadratic expression

By being aware of these common mistakes, we can ensure that we factor the trinomial correctly and avoid any errors.

Real-World Applications

Factoring a trinomial has numerous real-world applications in fields such as:

  • Algebra: Factoring a trinomial is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials.
  • Calculus: Factoring a trinomial is used in calculus to solve equations and find the derivative of a function.
  • Physics: Factoring a trinomial is used in physics to solve equations and find the velocity and acceleration of an object.

Tips and Tricks

Here are some tips and tricks to help you factor a trinomial:

  • Identify the GCF correctly by finding the largest factor that divides each term evenly.
  • Factor the quadratic expression correctly by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
  • Multiply the GCF with the factored quadratic expression to get the final factored form of the trinomial.

Conclusion

Introduction

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 provide a Q&A guide to help you understand the concept of factoring a trinomial completely.

Q: What is a trinomial?

A: A trinomial is a polynomial with three terms. It is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials.

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

A: The GCF is the largest factor that divides each term of the trinomial evenly. It is essential to identify the GCF correctly to factor the trinomial completely.

Q: How do I identify the GCF?

A: To identify the GCF, you need to find the largest factor that divides each term of the trinomial evenly. You can do this by listing the factors of each term and finding the common factors.

Q: What is the quadratic expression?

A: The quadratic expression is a polynomial with two terms. It is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials.

Q: How do I factor the quadratic expression?

A: To factor the quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. You can then write the quadratic expression as a product of two binomials.

Q: What is the final factored form of the trinomial?

A: The final factored form of the trinomial is obtained by multiplying the GCF with the factored quadratic expression.

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

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

  • Not identifying the GCF correctly
  • Not factoring the quadratic expression correctly
  • Not multiplying the GCF with the factored quadratic expression

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

A: Factoring a trinomial has numerous real-world applications in fields such as:

  • Algebra: Factoring a trinomial is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials.
  • Calculus: Factoring a trinomial is used in calculus to solve equations and find the derivative of a function.
  • Physics: Factoring a trinomial is used in physics to solve equations and find the velocity and acceleration of an object.

Q: What are some tips and tricks to help me factor a trinomial?

A: Here are some tips and tricks to help you factor a trinomial:

  • Identify the GCF correctly by finding the largest factor that divides each term evenly.
  • Factor the quadratic expression correctly by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
  • Multiply the GCF with the factored quadratic expression to get the final factored form of the trinomial.

Conclusion

Factoring a trinomial completely involves identifying the GCF and factoring the quadratic expression. By following the step-by-step approach outlined in this article, you can factor the trinomial 10y3−25y2−125y10y^3 - 25y^2 - 125y completely. The final factored form of the trinomial is 5y(2y+5)(y−5)5y(2y + 5)(y - 5).