Can you please give a example so that I can understand it better.
1)How can you tell whether a polynomial is written in factor form.
2)Construct a trinomial whose greatest common factor is 2x^2.
Sure! Here are step-by-step explanations for each of your questions:
1) How can you tell whether a polynomial is written in factor form?
Step 1: Look at the polynomial expression.
Step 2: Check if it can be factored by determining if there are any common factors that can be taken out from every term.
Step 3: If there are common factors in every term, then the polynomial is written in factor form.
Let's take an example to understand this better:
Given polynomial: 3x^2 + 6x
Step 1: Look at the expression, which is 3x^2 + 6x.
Step 2: Each term has a common factor of 3x. We can factor it out like this: 3x(x + 2).
Step 3: Since all the terms share the common factor, it means that the polynomial is written in factor form.
2) Construct a trinomial whose greatest common factor is 2x^2.
Step 1: Start with the greatest common factor, which is 2x^2.
Step 2: Add two variable terms, such as ax and bx, where a and b can be any coefficients.
Step 3: Combine all the terms together to form the trinomial.
The trinomial will be: 2x^2 + ax + bx
For example, if a = 3 and b = 4, then the trinomial will be:
2x^2 + 3x + 4x
Sure! Let's start with the first question.
1) How can you tell whether a polynomial is written in factor form?
To determine whether a polynomial is written in factor form, you need to check if it can be expressed as a product of its factors. Here's how you can do it:
Step 1: Identify the polynomial. For example, let's consider the polynomial 2x^2 - 8x + 12.
Step 2: Factor the polynomial, if possible. In this case, we can factor out the greatest common factor of the polynomial, which is 2: 2(x^2 - 4x + 6).
Step 3: Look for more potential factors. If the remaining expression in parentheses (x^2 - 4x + 6) is factorable, then the polynomial is written in factor form. In this case, we can further factor it to (x - 2)(x - 3). Therefore, the polynomial 2x^2 - 8x + 12 can be expressed in factor form as 2(x - 2)(x - 3).
So, by factoring the polynomial, you can determine whether it is written in factor form or not.
Now, let's move on to the second question.
2) Construct a trinomial whose greatest common factor is 2x^2.
To construct a trinomial with a greatest common factor of 2x^2, follow these steps:
Step 1: Start with the greatest common factor. In this case, the greatest common factor is 2x^2.
Step 2: Determine the remaining factors of the trinomial. For example, let's say we want the trinomial to have two more factors: (x + 3) and (x - 2).
Step 3: Multiply the greatest common factor with the remaining factors. Using the trinomial format, we multiply 2x^2 with (x + 3) and (x - 2) to get:
2x^2(x + 3)(x - 2)
So, the trinomial whose greatest common factor is 2x^2 is 2x^2(x + 3)(x - 2).
By following these steps, you can construct a trinomial with a specific greatest common factor.