Question

which polynomial is prime?
$3x^3 + 3x^2 - 2x - 2$
$3x^3 - 2x^2 + 3x - 4$
$4x^3 + 2x^2 + 6x + 3$
$4x^3 + 4x^2 - 3x - 3$

Explanation:

Step1: Factor Option 1 by grouping

Group terms: $(3x^3+3x^2)+(-2x-2)$
Factor out GCFs: $3x^2(x+1)-2(x+1)=(3x^2-2)(x+1)$

Step2: Factor Option 2 by grouping

Group terms: $(3x^3-2x^2)+(3x-4)$
Factor out GCFs: $x^2(3x-2)+1(3x-4)$
No common binomial factor. Test rational roots: Possible roots $\pm1,\pm2,\pm4,\pm\frac{1}{3},\pm\frac{2}{3},\pm\frac{4}{3}$. None satisfy $3x^3-2x^2+3x-4=0$.

Step3: Factor Option 3 by grouping

Group terms: $(4x^3+2x^2)+(6x+3)$
Factor out GCFs: $2x^2(2x+1)+3(2x+1)=(2x^2+3)(2x+1)$

Step4: Factor Option 4 by grouping

Group terms: $(4x^3+4x^2)+(-3x-3)$
Factor out GCFs: $4x^2(x+1)-3(x+1)=(4x^2-3)(x+1)$

Answer:

B. $3x^3 - 2x^2 + 3x - 4$