Factor the expression completely or state that the polynomial is prime. 3x^2-3x-90

To factor the expression \( 3x^2 - 3x - 90 \), we follow these steps:

### Step 1: Factor out the greatest common factor (GCF)
The GCF of \( 3x^2 \), \( -3x \), and \( -90 \) is 3. So, we factor 3 out:

\[
3(x^2 - x - 30)
\]

### Step 2: Factor the quadratic expression \( x^2 - x - 30 \)
We need to find two numbers that multiply to \(-30\) (the constant term) and add to \(-1\) (the coefficient of \(x\)).

The numbers are \(-6\) and \(5\), because:

\[
-6 \times 5 = -30 \quad \text{and} \quad -6 + 5 = -1
\]

### Step 3: Write the factored form
Now we can factor \(x^2 - x - 30\) as:

\[
(x - 6)(x + 5)
\]

### Step 4: Write the final factored expression
Including the 3 that we factored out initially, the fully factored form of the expression is:

\[
3(x - 6)(x + 5)
\]

Thus, the expression \( 3x^2 - 3x - 90 \) is factored as \( 3(x - 6)(x + 5) \), and it is not a prime polynomial.

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Would you like more details on any step or further clarification?

Here are 5 related questions to expand your understanding:
1. How do you determine the greatest common factor (GCF) in a polynomial?
2. What are the steps to factor quadratic polynomials?
3. How do you recognize prime polynomials?
4. What happens if a quadratic doesn't factor over the integers?
5. How does factoring relate to solving quadratic equations?

**Tip:** When factoring, always check for a GCF before attempting to factor further.

Factor the expression completely or state that the polynomial is prime. 3x^2 - 3x - 90

Learn how to factor the quadratic expression 3x^2 - 3x - 90 completely with a step-by-step guide. This problem focuses on key algebraic concepts like factoring quadratics and identifying the greatest common factor. Suitable for students in Grades 8-10.

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