Question

sort the polynomials according to whether they are prime or non - prime.
$6x^3 - 5x^2 + 2x - 14$
$12x^4 - 18x^3 + 8x^2 - 12$
$8x^3 + 9x^2 + 20x + 15$
$12x^4 - 3x + 4x + 1$
prime polynomials
non - prime polynomials

Explanation:

Step1: Factor first polynomial by grouping

Group terms: $(6x^3 - 5x^2) + (2x - 16) = x^2(6x - 5) + 2(x - 8)$. No common binomial factor, so it cannot be factored over integers.

Step2: Factor second polynomial by grouping

Group terms: $(12x^3 - 18x^2) + (8x - 12) = 6x^2(2x - 3) + 4(2x - 3) = (6x^2 + 4)(2x - 3) = 2(3x^2 + 2)(2x - 3)$. This can be factored, so it is non-prime.

Step3: Factor third polynomial by grouping

Group terms: $(6x^3 + 9x^2) + (10x + 15) = 3x^2(2x + 3) + 5(2x + 3) = (3x^2 + 5)(2x + 3)$. This can be factored, so it is non-prime.

Step4: Factor fourth polynomial by grouping

Group terms: $(12x^3 - 3x) + (4x + 1) = 3x(4x^2 - 1) + 1(4x + 1) = 3x(2x-1)(2x+1) + 1(4x+1)$. No common binomial factor, so it cannot be factored over integers.

Answer:

Prime Polynomials:

$6x^3 - 5x^2 + 2x - 16$
$12x^3 - 3x + 4x + 1$

Non-Prime Polynomials:

$12x^3 - 18x^2 + 8x - 12$
$6x^3 + 9x^2 + 10x + 15$