which polynomial is prime?\n$7x^{2}-35x + 2x-10$\n$9x^{3}+11x^{2}+3x - 33$\n$10x^{3}-15x^{2}+8x…
which polynomial is prime?\n$7x^{2}-35x + 2x-10$\n$9x^{3}+11x^{2}+3x - 33$\n$10x^{3}-15x^{2}+8x - 12$\n$12x^{4}+42x^{2}+4x^{2}+14$
Answer
Explanation:
Step1: Factor the first polynomial
Factor $7x^{2}-35x + 2x-10$ by grouping. Group the terms: $(7x^{2}-35x)+(2x - 10)$. Factor out the greatest - common factor from each group: $7x(x - 5)+2(x - 5)=(7x + 2)(x - 5)$.
Step2: Factor the second polynomial
Factor $9x^{3}+11x^{2}+3x - 33$ by grouping. Group the terms: $(9x^{3}+11x^{2})+(3x - 33)$. Factor out the GCF from each group: $x^{2}(9x + 11)+3(x - 11)$. This can be further factored as $(9x + 11)(x^{2}+3)$.
Step3: Factor the third polynomial
Factor $10x^{3}-15x^{2}+8x - 12$ by grouping. Group the terms: $(10x^{3}-15x^{2})+(8x - 12)$. Factor out the GCF from each group: $5x^{2}(2x - 3)+4(2x - 3)=(2x - 3)(5x^{2}+4)$.
Step4: Factor the fourth polynomial
Factor $12x^{4}+42x^{2}+4x^{2}+14$ by grouping. Group the terms: $(12x^{4}+42x^{2})+(4x^{2}+14)$. Factor out the GCF from each group: $6x^{2}(2x^{2}+7)+2(2x^{2}+7)=(2x^{2}+7)(6x^{2}+2)=2(2x^{2}+7)(3x^{2}+1)$.
Since the first, second, third, and fourth polynomials can be factored into non - trivial factors, they are not prime.