Explanation of Exercise 2.4 Question 4:

The question states that if x + 1 is a factor of the polynomial x3 + ax2 + bx + c, then show that a + b + c = -1.

Steps to prove the statement:

To prove this statement, we need to use the factor theorem which states that if x + k is a factor of a polynomial, then the polynomial will be equal to (x + k) multiplied by another polynomial.

Let us assume that x + 1 is a factor of the given polynomial x3 + ax2 + bx + c. Then we can write:

x3 + ax2 + bx + c = (x + 1) (px2 + qx + r)

where p, q, and r are constants to be determined.

Now, we need to expand the right-hand side of the above equation to find the values of p, q, and r. We get:

x3 + ax2 + bx + c = px3 + (p + q)x2 + (q + r)x + r

Equating the coefficients of like terms on both sides, we get:

p = 1
p + q = a
q + r = b
r = c

Substituting p = 1 and r = c in the second and third equations, we get:

1 + q = a
q + c = b

Adding these two equations, we get:

1 + 2q + c = a + b
1 + 2q + c = -1 (since x + 1 is a factor)

Therefore, a + b + c = -1, which proves the statement.

Conclusion:

Thus, we have proved that if x + 1 is a factor of the polynomial x3 + ax2 + bx + c, then a + b + c = -1.