What Is The Factorization Of The Trinomial Below? 3 X 3 + 6 X 2 − 24 X 3x^3 + 6x^2 - 24x 3 X 3 + 6 X 2 − 24 X A. 3 X ( X − 2 ) ( X − 4 3x(x-2)(x-4 3 X ( X − 2 ) ( X − 4 ] B. 3 X ( X − 2 ) ( X + 4 3x(x-2)(x+4 3 X ( X − 2 ) ( X + 4 ] C. 3 ( X 2 − 2 ) ( X + 4 3(x^2-2)(x+4 3 ( X 2 − 2 ) ( X + 4 ] D. 3 ( X − 2 ) ( X + 4 3(x-2)(x+4 3 ( X − 2 ) ( X + 4 ]

Understanding Trinomials and Factorization

A trinomial is a polynomial expression consisting of three terms. Factorization is the process of expressing a trinomial as a product of simpler expressions, known as factors. In this article, we will explore the factorization of the given trinomial, $3x^3 + 6x^2 - 24x$.

The Importance of Factorization

Factorization is a crucial concept in algebra, as it allows us to simplify complex expressions and solve equations more easily. By factoring a trinomial, we can identify its roots, which are the values of the variable that make the expression equal to zero. This is particularly useful in solving quadratic equations, which are equations of the form $ax^2 + bx + c = 0$.

The Given Trinomial

The given trinomial is $3x^3 + 6x^2 - 24x$. To factorize this expression, we need to find two binomials whose product is equal to the given trinomial. A binomial is a polynomial expression consisting of two terms.

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factorizing the trinomial is to identify the greatest common factor (GCF) of the three terms. The GCF is the largest expression that divides each term evenly. In this case, the GCF is $3x$, as it divides each term evenly.

Step 2: Factor Out the GCF

Once we have identified the GCF, we can factor it out of each term. This leaves us with a quadratic expression inside the parentheses. In this case, we have:

$$3x^3 + 6x^2 - 24x = 3x(x^2 + 2x - 8)$$

Step 3: Factor the Quadratic Expression

Now we need to factor the quadratic expression inside the parentheses. A quadratic expression is a polynomial expression of degree two, which means it has a squared variable. In this case, the quadratic expression is $x^2 + 2x - 8$. We can factor this expression by finding two numbers whose product is equal to the constant term (-8) and whose sum is equal to the coefficient of the linear term (2). These numbers are 4 and -2, as $4 \times (-2) = -8$ and $4 + (-2) = 2$.

Step 4: Write the Factored Form

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

$$3x^3 + 6x^2 - 24x = 3x(x + 4)(x - 2)$$

Conclusion

In this article, we have factorized the given trinomial, $3x^3 + 6x^2 - 24x$. We identified the greatest common factor (GCF) and factored it out of each term. We then factored the quadratic expression inside the parentheses and wrote the factored form of the trinomial. The correct factorization of the trinomial is $3x(x + 4)(x - 2)$.

Answer

The correct answer is:

• A. $3x(x-2)(x-4)$ is incorrect, as the correct factorization is $3x(x + 4)(x - 2)$.
• B. $3x(x-2)(x+4)$ is correct, as it matches the factored form of the trinomial.
• C. $3(x^2-2)(x+4)$ is incorrect, as the correct factorization is $3x(x + 4)(x - 2)$.
• D. $3(x-2)(x+4)$ is incorrect, as the correct factorization is $3x(x + 4)(x - 2)$.

The final answer is B. $3x(x-2)(x+4)$.

Understanding Trinomials and Factorization

A trinomial is a polynomial expression consisting of three terms. Factorization is the process of expressing a trinomial as a product of simpler expressions, known as factors. In this article, we will explore the factorization of the given trinomial, $3x^3 + 6x^2 - 24x$.

Q&A: Factorization of Trinomials

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

A: The greatest common factor (GCF) of the three terms in the given trinomial is $3x$, as it divides each term evenly.

Q: How do you factor out the GCF from the trinomial?

A: To factor out the GCF, we divide each term by the GCF. In this case, we have:

$$3x^3 + 6x^2 - 24x = 3x(x^2 + 2x - 8)$$

Q: How do you factor the quadratic expression inside the parentheses?

A: To factor the quadratic expression inside the parentheses, we need to find two numbers whose product is equal to the constant term (-8) and whose sum is equal to the coefficient of the linear term (2). These numbers are 4 and -2, as $4 \times (-2) = -8$ and $4 + (-2) = 2$.

Q: What is the factored form of the trinomial?

A: The factored form of the trinomial is $3x(x + 4)(x - 2)$.

Q: Why is factorization important in algebra?

A: Factorization is important in algebra because it allows us to simplify complex expressions and solve equations more easily. By factoring a trinomial, we can identify its roots, which are the values of the variable that make the expression equal to zero.

Q: How do you identify the roots of a trinomial?

A: To identify the roots of a trinomial, we set the factored form of the trinomial equal to zero and solve for the variable. In this case, we have:

$$3x(x + 4)(x - 2) = 0$$

We can solve for the variable by setting each factor equal to zero:

• $3x = 0 \Rightarrow x = 0$
• $x + 4 = 0 \Rightarrow x = -4$
• $x - 2 = 0 \Rightarrow x = 2$

Therefore, the roots of the trinomial are $x = 0, x = -4$, and $x = 2$.

Conclusion

In this article, we have explored the factorization of the given trinomial, $3x^3 + 6x^2 - 24x$. We identified the greatest common factor (GCF) and factored it out of each term. We then factored the quadratic expression inside the parentheses and wrote the factored form of the trinomial. We also answered some common questions about factorization and its importance in algebra.

Answer

The correct answer is:

• A. $3x(x-2)(x-4)$ is incorrect, as the correct factorization is $3x(x + 4)(x - 2)$.
• B. $3x(x-2)(x+4)$ is correct, as it matches the factored form of the trinomial.
• C. $3(x^2-2)(x+4)$ is incorrect, as the correct factorization is $3x(x + 4)(x - 2)$.
• D. $3(x-2)(x+4)$ is incorrect, as the correct factorization is $3x(x + 4)(x - 2)$.

The final answer is B. $3x(x-2)(x+4)$.