Understanding the Basics of Trinomial Factorization

Oct 13, 2025 by magicalceylontours.com 56 views

Understanding the Basics of Trinomial Factorization

Trinomial factorization is a mathematical process used to express a trinomial as a product of simpler expressions, known as factors. A trinomial is a polynomial with three terms, and factorizing it can help us solve equations, find roots, and simplify complex expressions. In this article, we will explore the factorization of a given trinomial and determine the correct factorization among the provided options.

The Given Trinomial

The trinomial we are given is:

${ x^3 - 12x^2 + 35x }$

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factorizing a trinomial is to identify the greatest common factor (GCF) of the three terms. In this case, the GCF is x, as it is the highest factor that divides all three terms.

Step 2: Factor Out the GCF

We can now factor out the GCF, x, from each term:

${ x(x^2 - 12x + 35) }$

Step 3: Factor the Quadratic Expression

The remaining quadratic expression, x^2 - 12x + 35, can be factored further. To do this, we need to find two numbers whose product is 35 and whose sum is -12. These numbers are -7 and -5, as (-7) × (-5) = 35 and (-7) + (-5) = -12.

Step 4: Write the Factored Form

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

${ x(x - 7)(x - 5) }$

Conclusion

Based on our step-by-step factorization process, we have determined that the correct factorization of the given trinomial is:

${ x(x - 7)(x - 5) }$

This matches option B in the provided choices.

Comparison with Other Options

Let's compare our result with the other options:

Option A: x(x - 7)(x + 5) is incorrect, as the quadratic expression x^2 - 12x + 35 does not factor to (x - 7)(x + 5).
Option C: (x^2 - 7)(x + 5) is incorrect, as the quadratic expression x^2 - 12x + 35 does not factor to (x^2 - 7)(x + 5).
Option D: (x - 7)(x + 5) is incorrect, as it does not account for the x term in the original trinomial.

Final Answer

The correct factorization of the given trinomial is:

${ x(x - 7)(x - 5) }$

This is option B in the provided choices.

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of the three terms in the trinomial?

A: The GCF is x, as it is the highest factor that divides all three terms.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term.

Q: What is the difference between a trinomial and a quadratic expression?

A: A trinomial is a polynomial with three terms, while a quadratic expression is a polynomial with two terms.

Additional Resources

For more information on trinomial factorization, see this article.
For practice problems and exercises, see this website.

Conclusion

In this article, we have explored the factorization of a given trinomial and determined the correct factorization among the provided options. We have also discussed the basics of trinomial factorization, including identifying the greatest common factor (GCF) and factoring quadratic expressions. By following these steps, you can factorize any trinomial and solve equations, find roots, and simplify complex expressions.

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of the three terms in the trinomial?

A: The GCF is the highest factor that divides all three terms. In the case of the trinomial x^3 - 12x^2 + 35x, the GCF is x, as it is the highest factor that divides all three terms.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. For example, in the quadratic expression x^2 - 12x + 35, you need to find two numbers whose product is 35 and whose sum is -12. These numbers are -7 and -5, as (-7) × (-5) = 35 and (-7) + (-5) = -12.

Q: What is the difference between a trinomial and a quadratic expression?

A: A trinomial is a polynomial with three terms, while a quadratic expression is a polynomial with two terms. A trinomial can be factored into the product of a linear expression and a quadratic expression, while a quadratic expression can be factored into the product of two binomials.

Q: How do I determine the correct factorization of a trinomial?

A: To determine the correct factorization of a trinomial, you need to follow these steps:

Identify the greatest common factor (GCF) of the three terms.
Factor out the GCF from each term.
Factor the remaining quadratic expression.
Write the factored form of the trinomial.

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

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

Not identifying the greatest common factor (GCF) of the three terms.
Factoring out the wrong factor from each term.
Not factoring the quadratic expression correctly.
Writing the factored form of the trinomial incorrectly.

Q: How do I check my work when factoring trinomials?

A: To check your work when factoring trinomials, you can multiply the factors together to see if you get the original trinomial. For example, if you factor the trinomial x^3 - 12x^2 + 35x into x(x - 7)(x - 5), you can multiply the factors together to get:

x(x - 7)(x - 5) = x(x^2 - 5x - 7x + 35) = x(x^2 - 12x + 35)

This shows that the factored form of the trinomial is correct.

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

A: Trinomial factorization has many real-world applications, including:

Solving equations and finding roots.
Simplifying complex expressions.
Factoring polynomials and finding their roots.
Solving systems of equations.

Additional Resources

For more information on trinomial factorization, see this article.
For practice problems and exercises, see this website.

Conclusion

In this article, we have answered some frequently asked questions about trinomial factorization. We have discussed the basics of trinomial factorization, including identifying the greatest common factor (GCF) and factoring quadratic expressions. We have also provided some common mistakes to avoid and some real-world applications of trinomial factorization. By following these steps and practicing with examples, you can become proficient in factoring trinomials and solving equations.