Which polynomial is prime?
x ^ 3 + 3x ^ 2 + 2x + 6
x ^ 3 + 3x ^ 2 – 2x – 6
10x ^ 2 – 4x + 3x + 6
10x ^ 2 – 10x + 6x – 6
The Correct Answer and Explanation is :
To determine which polynomial is prime, we need to check whether any of them can be factored into simpler polynomial expressions with integer coefficients. A polynomial is prime if it cannot be factored into two or more nontrivial polynomials with integer coefficients.
Step-by-Step Factorization:
- ( x^3 + 3x^2 + 2x + 6 )
- Grouping: ( (x^3 + 3x^2) + (2x + 6) )
- Factor: ( x^2(x + 3) + 2(x + 3) )
- Factor further: ( (x + 3)(x^2 + 2) )
- Not Prime (since it factors into ((x+3)(x^2+2)))
- ( x^3 + 3x^2 – 2x – 6 )
- Grouping: ( (x^3 + 3x^2) + (-2x – 6) )
- Factor: ( x^2(x + 3) – 2(x + 3) )
- Factor further: ( (x + 3)(x^2 – 2) )
- Not Prime (since it factors into ((x+3)(x^2-2)))
- ( 10x^2 – 4x + 3x + 6 )
- Simplify: ( 10x^2 – x + 6 )
- Check for factorization: There are no integer factors.
- Prime Polynomial
- ( 10x^2 – 10x + 6x – 6 )
- Simplify: ( 10x^2 – 4x – 6 )
- Factor: ( 2(5x^2 – 2x – 3) )
- Further factor: ( 2(5x + 3)(x – 1) )
- Not Prime (since it factors into ( 2(5x + 3)(x – 1) ))
Conclusion:
The only prime polynomial is ( 10x^2 – x + 6 ) because it cannot be factored further using integer coefficients.
Now, I will generate an image illustrating the factorization of these polynomials.
The image illustrates the step-by-step factorization of the given polynomials, highlighting that ( 10x^2 – x + 6 ) is prime because it cannot be factored further. Let me know if you need any modifications or further explanations!
