which polynomials are prime? check all that apply. 15x² + 10x - 9x + 7 20x² - 12x + 30x - 18 6x³ + 14x²…

Explanation:

Step1: Simplify and factor each polynomial

• For (15x^{2}+10x - 9x+7): Combine like - terms: (15x^{2}+(10x - 9x)+7=15x^{2}+x + 7). Using the discriminant formula (D=b^{2}-4ac) for (ax^{2}+bx + c) ((a = 15), (b = 1), (c = 7)), (D=1^{2}-4\times15\times7=1 - 420=-419<0). So it cannot be factored over the real numbers.
• For (20x^{2}-12x + 30x-18): Group terms: ((20x^{2}-12x)+(30x - 18)=4x(5x - 3)+6(5x - 3)=(5x - 3)(4x + 6)=2(5x - 3)(2x + 3)).
• For (6x^{3}+14x^{2}-12x - 28): Group terms: ((6x^{3}+14x^{2})+(-12x - 28)=2x^{2}(3x + 7)-4(3x + 7)=(3x + 7)(2x^{2}-4)=2(3x + 7)(x^{2}-2)).
• For (8x^{3}+20x^{2}+3x + 12): Group terms: ((8x^{3}+20x^{2})+(3x + 12)=4x^{2}(2x + 5)+3(2x + 5)=(2x + 5)(4x^{2}+3)).
• For (11x^{4}+4x^{2}-6x^{2}-16): Combine like - terms: (11x^{4}+(4x^{2}-6x^{2})-16=11x^{4}-2x^{2}-16). Let (y = x^{2}), then the polynomial becomes (11y^{2}-2y - 16). Using the discriminant formula (D=b^{2}-4ac) ((a = 11), (b=-2), (c=-16)), (D=(-2)^{2}-4\times11\times(-16)=4 + 704 = 708=4\times177). (y=\frac{2\pm\sqrt{708}}{22}=\frac{2\pm2\sqrt{177}}{22}=\frac{1\pm\sqrt{177}}{11}). Since (\sqrt{177}) is irrational, (11x^{4}-2x^{2}-16) cannot be factored over the integers.

Answer:

(15x^{2}+x + 7) and (8x^{3}+20x^{2}+3x + 12)