Why are a primitive polynomial equations used in linear feedback shift registers?
Providing specific examples from the text may be easier to address. 

But something similar is done for CRCs. Polynomials are occasionally used to express the values in certain bit positions. In your example, \$x^4+x+1\$ would be 10011.

If you fill in the zeros for the polynomial this makes more sense: 

$$1x^4+0x^3+0x^2+1x^1+1x^0$$

The coefficients are the bit value and the rest is the value position. This can sometimes save a some writing and confusion if you had a large binary number like: 

$$10000000000000001000000000000000 $$

This would simply be written in polynomial form with: $$x^{31}+x^{15}$$

For the LSFRs it's defining which bit positions are being used to generate the next value of the register. It's much easier to provide the polynomial defining the bit positions than it is to provide a binary number where the 1s are the bit positions.