Question

find the constant of the polynomial (a+3)(x^b-1)-(a+2b)(b-1) if it is monic and of degree 4. the answer should be -19.

Answer 1

Given that the polynomial is monic and of degree 4, we know that the leading coefficient is 1.

Expanding the polynomial (a+3)(x^b-1)-(a+2b)(b-1), we get:

(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 - 2b^2 - 3.

Since the polynomial is monic and of degree 4, the term with x^4 must have a coefficient of 1. Therefore, we solve for a and b:

ax^b = x^4
=> ab = 4
3x^b - 2b^2 = x^4 - 3
=> 3 = 1
=> b = 1

Now substitute b = 1 into ab = 4:
a = 4/1 = 4

Substitute a = 4 and b = 1 back into the expression:
ax^b + 3x^b - 2b^2 - 3
= 4x + 3 - 2 - 3
= 4x - 2

Therefore, the constant term of the polynomial is -2, not -19. It seems there may be an error in the previous calculation. If you could provide additional information or clarify the problem, I would be happy to assist further.