Which polynomial is prime? X3 + 3×2 + 2x + 6 O x3 + 3×2 – 2x – 6 0 10×2 – 4x + 3x + 6 10×2 – 10x + 6x – 6
The Correct Answer and Explanation is :
To determine which polynomial is prime, we must check whether any of the given polynomials can be factored into the product of two or more nontrivial polynomials with integer coefficients.
Step 1: Analyze Each Polynomial
- ( x^3 + 3x^2 + 2x + 6 )
- Group terms: ( (x^3 + 3x^2) + (2x + 6) ).
- Factor partially: ( x^2(x + 3) + 2(x + 3) ).
- Factor completely: ( (x^2 + 2)(x + 3) ).
- Not prime because it can be factored.
- ( x^3 + 3x^2 – 2x – 6 )
- Group terms: ( (x^3 + 3x^2) + (-2x – 6) ).
- Factor partially: ( x^2(x + 3) – 2(x + 3) ).
- Factor completely: ( (x^2 – 2)(x + 3) ).
- Not prime because it can be factored.
- ( 10x^2 – 4x + 3x + 6 )
- Simplify: ( 10x^2 – x + 6 ).
- Check if factorable: It has no rational factors.
- Prime Polynomial.
- ( 10x^2 – 10x + 6x – 6 )
- Simplify: ( 10x^2 – 4x – 6 ).
- Check if factorable: ( 10x^2 – 4x – 6 = (2x – 2)(5x + 3) ).
- Not prime because it can be factored.
Conclusion:
The prime polynomial is ( 10x^2 – x + 6 ) because it cannot be factored into simpler polynomials with integer coefficients.
Now, I will generate an image representing this analysis.
Here is the generated image displaying the polynomial equations and their factored forms, highlighting the prime polynomial (10x^2 – x + 6). Let me know if you need any modifications or further explanations!
