Question

which polynomials are prime? check all that apply. \\(15x^2 + 10x - 9x + 7\\) \\(20x^2 - 12x + 30x - 18\\) \\(6x^3 + 14x^2 - 12x - 28\\) \\(8x^3 + 20x^2 + 3x + 12\\) \\(11x^4 + 4x^2 - 6x^2 - 16\\)

Explanation:

Step1: Simplify & factor first polynomial

Combine like terms: $15x^2 + (10x-9x) +7 = 15x^2+x+7$.
Check discriminant: $\Delta = 1^2 -4(15)(7)=1-420=-419<0$. No real factors, so prime.

Step2: Factor second polynomial

Group terms: $(20x^2-12x)+(30x-18)=4x(5x-3)+6(5x-3)=(4x+6)(5x-3)$. Not prime.

Step3: Factor third polynomial

Group terms: $(6x^3+14x^2)+(-12x-28)=2x^2(3x+7)-4(3x+7)=(2x^2-4)(3x+7)$. Not prime.

Step4: Simplify & factor fourth polynomial

Group terms: $(8x^3+20x^2)+(3x+12)=4x^2(2x+5)+3(x+4)$. No common binomial factor. Check discriminant if treated as quadratic in $x$: no rational/real factors, so prime.

Step5: Simplify & factor fifth polynomial

Combine like terms: $11x^4 + (4x^2-6x^2)-16=11x^4-2x^2-16$. Let $y=x^2$: $11y^2-2y-16$. Factor: $(11y+16)(y-1)=(11x^2+16)(x^2-1)=(11x^2+16)(x-1)(x+1)$. Not prime.

Answer:

$15x^2 + 10x - 9x + 7$, $8x^3 + 20x^2 + 3x + 12$