Question

Which polynomial is prime? 3x3 + 3x2 – 2x – 2 3x3 – 2x2 + 3x – 4 4x3 + 2x2 + 6x + 3 4x3 + 4x2 – 3x – 3

Answer 1

Consider all polynomials:

1. $3x^3 + 3x^2 - 2x - 2=(3x^3+3x^2)+(-2x-2)=3x^2(x+1)-2(x+1)=(x+1)(3x^2-2),$ this polynomial is not prime because you can factor it as shown above.

3. $4x^3 + 2x^2 + 6x + 3=(4x^3 + 2x^2) + (6x + 3)=2x^2(2x+1)+3(2x+1)=(2x+1)(2x^2+3),$ this polynomial is not prime because you can factor it as shown above.

4. $4x^3 + 4x^2 - 3x - 3=4x^3 + 4x^2 +(- 3x - 3)=4x^2(x+1)-3(x+1)=(x+1)(4x^2-3),$ this polynomial is not prime because you can factor it as shown above.

2. Only this polynomial cannot be factored.

Answer: correct option is 2.

Answer 2

Answer:$3x^3-2x^2+3x-4$

Step-by-step explanation:

We know that a polynomial is said to be prime if it does not factorize into polynomials.

1. $3x^3+3x^2-2x-2\\=3x^2(x+1)-1(x+1)\\=(x+1)(3x^2-1)$

This is not prime as it can factorize.

2.$3x^3-2x^2+3x-4\\=x^2(3x-2)+1(3x-4)$

$(3x-2)\ and\ (3x-4)$ are not same , thus we cannot factorize the above polynomial.

Therefore this polynomial is prime.

3.$4x^3+2x^2+6^x+3=2x^2(2x+1)+3(2x+1)\\=(2x+1)(2x^2+3)$

This is not prime as it can factorize.

4.$4x^3+4x^2-3x-3\\=4x^2(x+1)-3(x+1)\\=(x+1)(4x^2-3)$

This is not prime as it can factorize.