Which polynomial is prime?

x3 + 3x2 + 2x + 6
x3 + 3x2 – 2x – 6
10x2 – 4x + 3x + 6
10x2 – 10x + 6x – 6

To determine which polynomial is prime, we need to check if any of these polynomials can be factored into lower-degree polynomials with integer coefficients. A polynomial is considered prime if it cannot be factored further.

Let's analyze each polynomial:

1. \( x^3 + 3x^2 + 2x + 6 \)

   We can use the Rational Root Theorem to test possible rational roots (\(\pm 1, \pm 2, \pm 3, \pm 6\)). None of these values satisfy the polynomial, suggesting it might be prime. For a cubic polynomial, a failure of the Rational Root Test indicates primality, but we can verify this by attempting factorization, which is complex and beyond simple rational roots. Let’s leave it and check others first for easier cases.

2. \( x^3 + 3x^2 - 2x - 6 \)

   Using the Rational Root Theorem to test possible rational roots (\(\pm 1, \pm 2, \pm 3, \pm 6\)), none of these values satisfy the polynomial, suggesting this might also be prime. Let’s also leave it and check others.

3. \( 10x^2 - 4x + 3x + 6 \)

   Combine like terms to simplify:
   \[
   10x^2 - x + 6
   \]
   Use the discriminant method to check factorability:
   \[
   \Delta = (-1)^2 - 4 \cdot 10 \cdot 6 = 1 - 240 = -239
   \]
   Since the discriminant is negative, this quadratic polynomial has no real roots and hence is prime over the real numbers.

4. \( 10x^2 - 10x + 6x - 6 \)

   Combine like terms to simplify:
   \[
   10x^2 - 4x - 6
   \]
   Use the discriminant method to check factorability:
   \[
   \Delta = (-4)^2 - 4 \cdot 10 \cdot (-6) = 16 + 240 = 256
   \]
   Since the discriminant is positive, this quadratic polynomial can be factored.

Given the above analysis, we can conclude that \( 10x^2 - x + 6 \) is a prime polynomial as it cannot be factored into polynomials of lower degree with integer coefficients.

Would you like more details on the steps, or do you have any questions?

Here are five related questions you can ask:
1. How can you factorize a polynomial using the Rational Root Theorem?
2. What is the discriminant and how does it determine the nature of polynomial roots?
3. Can you provide more examples of prime polynomials?
4. How do you factor a cubic polynomial?
5. What are the steps to simplify polynomials before factorization?

**Tip:** Always simplify the polynomial first before attempting to factorize it, as it can often reveal hidden factors more easily.

Learn how to determine if a polynomial is prime with a detailed explanation and step-by-step analysis of polynomials. This guide covers key concepts such as the Rational Root Theorem and the discriminant method for quadratic equations. Perfect for high school students and algebra enthusiasts.

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