Respuesta :

frika

1.

[tex] 7x^2-35x + 2x - 10=7x^2-33x-10,\\ D=(-33)^-4\cdot 7\cdot (-10)=1369,\\ \sqrt{D}=37,\\ \\ x_1=\dfrac{33-37}{2\cdot 7}=-\dfrac{2}{7},\ x_2=\dfrac{33+37}{2\cdot 7}=5,\\ \\ 7x^2-35x + 2x - 10=7x^2-33x-10=7(x+\dfrac{2}{7})(x-5)=(7x+2)(x-5) [/tex] - this polynomial is not prime.

3.

[tex] 10x^3 - 15x^2 + 8x- 12=(10x^3-15x^2)+(8x-12)=5x^2(2x-3)+4(2x-3)=(2x-3)(5x^2+4) [/tex] - this polynomial is not prime.

4.

[tex] 12x^4 + 42x^2 + 4x^2 + 14=12x^4+46x^2+14=2(6x^4+23x^2+7),\\D=23^2-4\cdot 6\cdot 7=361, \sqrt{D}=19,\\ \\x_1^2=\dfrac{-23-19}{12} =-\dfrac{7}{2} , x_2^2=\dfrac{-23+19}{12}=-\dfrac{1}{3} ,\\ \\ 12x^4 + 42x^2 + 4x^2 + 14=2(6x^4+23x^2+7)=12(x^2+\dfrac{7}{2})(x^2+\dfrac{1}{3}) [/tex] - this polynomial is not prime.

2. This polynomial is prime.

Answer:

Option 2 is correct.

Step-by-step explanation:

Given the polynomials we have to choose the polynomial which is prime.

A polynomial with integer coefficients that cannot be factored into lower degree polynomials are Prime polynomials.

[tex]7x^2 - 35x + 2x - 10[/tex]

⇒ [tex]7x(x-5)+2(x-5)[/tex]

⇒ [tex](7x+2)(x-5)[/tex]

can be factored ∴ not a prime polynomial.

[tex]9x^3 + 11x^2 + 3x - 33[/tex]

⇒ [tex]x^2(9x+11)+3(x-11)[/tex]

cannot be factored ∴ prime polynomial.

[tex]10x^3 - 15x^2 + 8x - 12[/tex]

⇒ [tex]5x^2(2x-3)+4(2x-3)[/tex]

⇒ [tex](5x^2+4)(2x-3)[/tex]

can be factored ∴ not a prime polynomial.

[tex]12x^4 + 42x^2 + 4x^2 + 14[/tex]

⇒ [tex]6x^2(2x^2+7)+2(2x^2+7)[/tex]

⇒ [tex](6x^2+2)(2x^2+7)[/tex]

can be factored ∴ not a prime polynomial.

The second polynomial can not be factored into lower degree polynomial therefore, prime polynomial.