Which Is The Completely Factored Form Of $3x^2 - 12x - 15$?A. $3(x + 1)(x - 5$\]B. $3(x - 1)(x + 5$\]C. $(3x + 1)(x - 15$\]

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of simpler expressions. In this article, we will explore the process of factoring quadratic expressions and apply it to the given quadratic expression $3x^2 - 12x - 15$. We will examine the different options provided and determine which one is the completely factored form of the given expression.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has the highest power of the variable (in this case, x) equal to two. The general form of a quadratic expression is $ax^2 + bx + c$, where a, b, and c are constants. In the given expression $3x^2 - 12x - 15$, a = 3, b = -12, and c = -15.

Factoring Quadratic Expressions

Factoring a quadratic expression involves expressing it as a product of two binomials. The process of factoring a quadratic expression can be broken down into several steps:

  1. Identify the greatest common factor (GCF): The first step in factoring a quadratic expression is to identify the greatest common factor (GCF) of the terms. In the given expression $3x^2 - 12x - 15$, the GCF is 3.
  2. Factor out the GCF: Once the GCF is identified, we can factor it out of the expression. This involves dividing each term by the GCF and writing the result as a product of the GCF and the remaining terms.
  3. Identify the remaining quadratic expression: After factoring out the GCF, we are left with a quadratic expression that can be factored further.
  4. Use the quadratic formula or other factoring techniques: If the remaining quadratic expression cannot be factored further, we can use the quadratic formula or other factoring techniques to factor it.

Factoring the Given Expression

Now that we have a general understanding of the factoring process, let's apply it to the given expression $3x^2 - 12x - 15$. The first step is to identify the GCF, which is 3. We can factor out the GCF by dividing each term by 3:

3x2−12x−15=3(x2−4x−5)3x^2 - 12x - 15 = 3(x^2 - 4x - 5)

Next, we need to factor the remaining quadratic expression $x^2 - 4x - 5$. We can use the quadratic formula or other factoring techniques to factor it. In this case, we can factor it as:

x2−4x−5=(x−5)(x+1)x^2 - 4x - 5 = (x - 5)(x + 1)

Therefore, the completely factored form of the given expression $3x^2 - 12x - 15$ is:

3(x−5)(x+1)3(x - 5)(x + 1)

Conclusion

In this article, we explored the process of factoring quadratic expressions and applied it to the given expression $3x^2 - 12x - 15$. We identified the GCF, factored it out, and then factored the remaining quadratic expression using the quadratic formula or other factoring techniques. The completely factored form of the given expression is $3(x - 5)(x + 1)$. This result is consistent with option A, which is the correct answer.

Comparison of Options

Let's compare the options provided to determine which one is the completely factored form of the given expression.

  • Option A: 3(x+1)(x−5)3(x + 1)(x - 5)
  • Option B: 3(x−1)(x+5)3(x - 1)(x + 5)
  • Option C: (3x+1)(x−15)(3x + 1)(x - 15)

Option A is the correct answer, as it matches the completely factored form of the given expression $3(x - 5)(x + 1)$.

Final Thoughts

Introduction

In our previous article, we explored the process of factoring quadratic expressions and applied it to the given expression $3x^2 - 12x - 15$. We identified the GCF, factored it out, and then factored the remaining quadratic expression using the quadratic formula or other factoring techniques. In this article, we will provide a Q&A guide to help you better understand the process of factoring quadratic expressions.

Q: What is the greatest common factor (GCF) of a quadratic expression?

A: The greatest common factor (GCF) of a quadratic expression is the largest expression that divides each term of the expression without leaving a remainder.

Q: How do I identify the GCF of a quadratic expression?

A: To identify the GCF of a quadratic expression, you need to find the largest expression that divides each term of the expression without leaving a remainder. You can do this by listing the factors of each term and finding the common factors.

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

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while simplifying a quadratic expression involves combining like terms to reduce the expression to its simplest form.

Q: Can a quadratic expression be factored if it has no GCF?

A: Yes, a quadratic expression can be factored even if it has no GCF. In this case, you can use other factoring techniques, such as the quadratic formula or completing the square, to factor the expression.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to factor a quadratic expression?

A: To use the quadratic formula to factor a quadratic expression, you need to first identify the values of a, b, and c in the expression. Then, you can plug these values into the quadratic formula to find the solutions to the equation.

Q: What are some common factoring techniques used to factor quadratic expressions?

A: Some common factoring techniques used to factor quadratic expressions include:

  • Factoring by grouping: This involves grouping the terms of the expression into pairs and factoring out the common factors from each pair.
  • Factoring by difference of squares: This involves factoring an expression of the form $a^2 - b^2$ as $(a + b)(a - b)$.
  • Factoring by sum and difference: This involves factoring an expression of the form $a^2 + b^2$ as $(a + b)(a - b)$.

Q: Can a quadratic expression be factored if it has a complex solution?

A: Yes, a quadratic expression can be factored even if it has a complex solution. In this case, you can use the quadratic formula to find the complex solutions to the equation.

Conclusion

In this article, we provided a Q&A guide to help you better understand the process of factoring quadratic expressions. We covered topics such as identifying the GCF, factoring by grouping, and using the quadratic formula to factor a quadratic expression. We also discussed some common factoring techniques used to factor quadratic expressions. By following these tips and techniques, you can become proficient in factoring quadratic expressions and solve a wide range of algebraic problems.

Final Thoughts

Factoring quadratic expressions is an essential skill in algebra that involves expressing a quadratic expression as a product of simpler expressions. By mastering the techniques and tips outlined in this article, you can become proficient in factoring quadratic expressions and solve a wide range of algebraic problems. Remember to always identify the GCF, factor by grouping, and use the quadratic formula to factor a quadratic expression. With practice and patience, you can become a master of factoring quadratic expressions.