Which polynomial is prime? a.) x2 + x + 1 b.) x2 – x – 2 c.) x2 – 12 d.) x + 11 x2 + 2x – 8

A) x²+x+1

Step-by-step explanation:

If a polynomial is prime, it cannot be factored.

The first polynomial, x²+x+1, cannot be factored; there are no two numbers whose product is 1 and whose sum is 1.

The second polynomial, x²-x-2, can be factored.  The numbers -2 and 1 have a product of -2 and a sum of 1; this makes the factored form

(x-2)(x+1)

The third polynomial can also be factored.  This is a difference of squares; the factors are found by taking the square root of each term:

(x+√12)(x-√12)

We can also simplify √12: √(4*3) = √4 * √3 = 2√3

This gives us

(x+2√3)(x-2√3)

The last polynomial can be factored by first combining like terms:

x+11x²+2x-8

x+2x+11x²-8

3x+11x²-8

11x²+3x-8

We want two numbers whose product is 11(-8) = -88 and whose sum is 3.  11 and -8 will work.  We split up b, the coefficient of x, using this:

11x²+11x-8x-8

Break this into two groups, separated by addition:

(11x²+11x)+(-8x-8)

Factor out the GCF of each group:

11x(x+1)+-8(x+1)

We have a common factor in each piece now, x+1:

(11x-8)(x+1)

The only one that could not be factored was A.