find the constant of the polynomial (a+3)(x^b-1)-(a+2b)(b-1) if it is monic and of degree 4.

Answer 1

First, let's expand the polynomial:
(a + 3)(x^b - 1) - (a + 2b)(b - 1)
= a(x^b - 1) + 3(x^b - 1) - a(b - 1) - 2b(b - 1)
= ax^b - a + 3x^b - 3 - ab + a - 2b^2 + 2b
= ax^b + 3x^b - 3 - ab - 2b^2 + 2b

If this polynomial is monic and of degree 4, then the leading term should be x^4. Therefore, the exponent of x in the highest-degree term is 4. This means b = 4.

Substitute b = 4 into the polynomial:
ax^4 + 3x^4 - 3 - 4a - 32 + 8
= 4x^4 + (4 - 4a - 24)

Since the polynomial is monic, the constant term should be 1. Therefore, the constant of the polynomial is 4 - 4a - 24 = 1.

4 - 4a - 24 = 1
-4a - 20 = 1
-4a = 21
a = -21/4

So, the constant of the polynomial is -21/4.