How do we find the greatest common factor of a polynomial? Demonstrate the process with an example, showing your work. When finding the greatest common factor of a polynomial, can it ever be larger than the smallest coefficient? Can it ever be smaller than the smallest coefficient?

• Explain in your own word how factoring is used to solve quadratic equations. Demonstrate the process with an example.

To find the greatest common factor of a polynomial, we can use the method of factoring. The steps for finding the greatest common factor are as follows:

1. First, factor each term in the polynomial completely.
2. Identify the common factors among all the terms.
3. Take the product of the common factors to find the greatest common factor.

Let's demonstrate the process with the polynomial example: 3x^3 + 6x^2 + 9x.

Step 1: Factor each term:
3x^3 can be factored as 3 * x * x * x
6x^2 can be factored as 2 * 3 * x * x
9x can be factored as 3 * 3 * x

Step 2: Identify the common factors:
The common factors among all the terms are 3 and x.

Step 3: Take the product of the common factors:
The greatest common factor of the polynomial is 3x.

Now, coming to your second question, can the greatest common factor ever be larger or smaller than the smallest coefficient? The answer is no. The greatest common factor will always be a factor of all the terms in the given polynomial. Therefore, it cannot be larger than or smaller than the smallest coefficient.

Now, let's move on to explaining how factoring is used to solve quadratic equations.

Factoring is a method used to solve quadratic equations by breaking down the equation into simpler terms. The goal is to express the quadratic equation in the form: (x + a)(x + b) = 0, where a and b are constants.

Here's an example to demonstrate the factoring process:

Let's solve the quadratic equation: x^2 + 5x + 6 = 0

Step 1: Arrange the equation in the form ax^2 + bx + c = 0:
x^2 + 5x + 6 = 0

Step 2: Factor the quadratic expression:
We need to find two numbers whose sum is 5 and product is 6. In this case, the numbers are 2 and 3.
(x + 2)(x + 3) = 0

Step 3: Set each factor equal to zero and solve for x:
x + 2 = 0 or x + 3 = 0

Solving the equations gives us:
x = -2 or x = -3

So the solutions to the quadratic equation x^2 + 5x + 6 = 0 are x = -2 and x = -3.

In summary, factoring is a method used to solve quadratic equations by breaking them down into simpler expressions, and the solutions are obtained by setting the factored terms equal to zero and solving for x.