Question

Which polynomial is prime?

A. 7x^2 – 35x + 2x – 10
B. 9x^3 + 11x^2 + 3x – 33
C. 10x^3 – 15x^2 + 8x – 12
D. 12x^4 + 42x^2 + 4x^2 + 14

Answer 1

B not C 9x^3 + 11x^2 + 3x - 33

Explanation:

Answer 2

C.
`9x^(3)+11 x^(2) +3x-33`

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.