Answer:
C.\(9x^{3}+11 x^{2} +3x-33\)
Step-by-step explanation:
A polynomial is prime if it can't be factored in polynomials of lower degree. Let's factorize:
A. \(7x^{2} -35x+2x-10\)
In this case we have 4 terms, so we can use Grouping:
Part a:
\(7x^{2} -35x\)
We're going to use Greatest common factor:
\(=7x^{2} -7.5x\\=7x(x-5)\)
Part b:
\(2x-10\)
In this part we also use greatest common factor:
\(2x-10=2x-2.5\\=2(x-5)\)
Then,
\(7x^{2} -35x+2x-10=\\ 7x(x-5)+2(x-5)=\\ =(7x+2)(x-5)\)
This polynomial is not prime.
B. \(9x^3 + 11x^2 + 3x-33\)
This polynomial cannot be factorized then it's prime.
C.\(10x^3-15x^2 + 8x-12\)
In this polynomial we can use grouping too:
Part a:
\(10x^3 -15x^2=2.5x^3-3.5x^2\\=5x^2(2x-3)\)
Part b:
\(8x-12=4.2x-4.3\\=4(2x-3)\)
Then,
\(10x^3-15x^2 + 8x-12=\\=5x^2(2x-3)+4(2x-3)\\=(5x^2+4)(2x-3)\)
This polynomial isn't prime.
D. \(12x^4 + 42x^2 + 4x^2 + 14\)
First we're going to use Greatest common factor:
\(12x^4 + 42x^2 + 4x^2 + 14=\\2.6x^4+2.21x^2+2.2x^2+2.7=2(6x^4+21x^2+2x^2+7)\)
Now we're going to apply grouping on the terms inside of the parenthesis:
\(6x^4+21x^2+2x^2+7\)
Part a:
\(6x^4+2x^2=2.3x^4+2x^2=2x^2(3x^2+1)\)
Part b:
\(21x^2+7=7.3x^2+7=7(3x^2+1)\)
Then,
\(6x^4+21x^2+2x^2+7=2x^2(3x^2+1)+7(3x^2+1)\\=(2x^2+7)(3x^2+1)\)
Remember that at the beginning we use Greatest common factor:
\(12x^4 + 42x^2 + 4x^2 + 14=\\=2.(6x^4+21x^2+2x^2+7)\\=2(2x^2+7)(3x^2+1)\)
This polynomial isn't prime.
