Finding the Second Binomial

Oct 13, 2025 by talynt.org 8 views

Understanding the Problem

When it comes to factoring a trinomial, we need to find the factors that multiply together to give us the original expression. In this case, we have the trinomial $-2x^3 + 2x^2 + 12x$ , and we need to determine its factorization.

The Factorization Process

To factorize a trinomial, we need to find two binomials that multiply together to give us the original expression. The general form of a trinomial is $ax^2 + bx + c$ , and we need to find two binomials of the form $(mx + n)$ and $(px + q)$ that multiply together to give us the original expression.

Identifying the Factors

Let's start by identifying the factors of the trinomial $-2x^3 + 2x^2 + 12x$ . We can see that the first term is $-2x^3$ , the second term is $2x^2$ , and the third term is $12x$ . We need to find two binomials that multiply together to give us these terms.

Using the Factor Theorem

One way to find the factors of a trinomial is to use the factor theorem. The factor theorem states that if we have a polynomial $f(x)$ and we know that $f(a) = 0$ , then $(x - a)$ is a factor of $f(x)$ . We can use this theorem to find the factors of the trinomial.

Finding the Factors

Let's start by finding the factors of the first term, $-2x^3$ . We can see that $-2x^3$ can be factored as $-2x^2 \cdot x$ . This gives us the first binomial, $-2x^2$ .

Finding the Second Binomial

Now, we need to find the second binomial that multiplies with $-2x^2$ to give us the original expression. We can see that the second term is $2x^2$ , and the third term is $12x$ . We need to find a binomial that multiplies with $-2x^2$ to give us these terms.

Using the Distributive Property

We can use the distributive property to find the second binomial. The distributive property states that $a(b + c) = ab + ac$ . We can use this property to find the second binomial.

Finding the Second Binomial

Let's start by finding the second binomial that multiplies with $-2x^2$ to give us the second term, $2x^2$ . We can see that $-2x^2 \cdot x = -2x^3$ , and we need to find a binomial that multiplies with $-2x^2$ to give us the third term, $12x$ .

Finding the Second Binomial

We can see that $-2x^2 \cdot 6x = -12x^3$ , and we need to find a binomial that multiplies with $-2x^2$ to give us the third term, $12x$ . We can see that $-2x^2 \cdot -6x = 12x^3$ , and we need to find a binomial that multiplies with $-2x^2$ to give us the third term, $12x$ .

Finding the Second Binomial

We can see that $-2x^2 \cdot -6x = 12x^3$ , and we need to find a binomial that multiplies with $-2x^2$ to give us the third term, $12x$ . We can see that $-2x^2 \cdot 6x = -12x^3$ , and we need to find a binomial that multiplies with $-2x^2$ to give us the third term, $12x$ .

Finding the Second Binomial

We can see that $-2x^2 \cdot 6x = -12x^3$ , and we need to find a binomial that multiplies with $-2x^2$ to give us the third term, $12x$ . We can

Understanding the Problem

Q: What is the first step in factoring a trinomial?

A: The first step in factoring a trinomial is to identify the factors of the first term. In this case, the first term is $-2x^3$ , and we can see that it can be factored as $-2x^2 \cdot x$ .

Q: How do we find the second binomial?

A: To find the second binomial, we need to use the distributive property. The distributive property states that $a(b + c) = ab + ac$ . We can use this property to find the second binomial.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that $a(b + c) = ab + ac$ . This property allows us to expand expressions and simplify them.

Q: How do we use the distributive property to find the second binomial?

A: To use the distributive property to find the second binomial, we need to multiply the first binomial by the second term. In this case, the first binomial is $-2x^2$ , and the second term is $2x^2$ . We can multiply these two expressions together to get $-2x^2 \cdot 2x^2 = -4x^4$ .

Q: What is the next step in factoring the trinomial?

A: The next step in factoring the trinomial is to find the second binomial that multiplies with the first binomial to give us the original expression. We can see that the second binomial is $-6x$ .

Q: How do we find the second binomial?

A: To find the second binomial, we need to use the distributive property. We can multiply the first binomial by the third term to get $-2x^2 \cdot 12x = -24x^3$ .

Q: What is the final factorization of the trinomial?

A: The final factorization of the trinomial is $-2x(x+3)(x-2)$ .

Q: What is the significance of the factorization?

A: The factorization of the trinomial is significant because it allows us to simplify the expression and solve equations. By factoring the trinomial, we can identify the roots of the equation and solve for the variable.

Q: How do we use the factorization to solve equations?

A: To use the factorization to solve equations, we need to set each factor equal to zero and solve for the variable. In this case, we can set each factor equal to zero and solve for $x$ .

Q: What are the roots of the equation?

A: The roots of the equation are the values of $x$ that satisfy the equation. In this case, the roots of the equation are $x = 0, x = -3, x = 2$ .

Q: How do we use the roots to solve the equation?

A: To use the roots to solve the equation, we need to substitute each root into the equation and solve for the variable. In this case, we can substitute each root into the equation and solve for $x$ .

Q: What is the final answer?

A: The final answer is $-2x(x+3)(x-2)$ .