When we divide the polynomial f(x) = 4x^3 + 3x^2 + ax + b by x – 1, we want to know if x – 1 is a factor of the polynomial. By the Factor Theorem, x – 1 is a factor of f(x) if f(1) = 0.
Let’s substitute x = 1 into the polynomial:
f(1) = 4(1)^3 + 3(1)^2 + a(1) + b
This simplifies to:
f(1) = 4 + 3 + a + b = 7 + a + b
For x – 1 to be a factor, we need:
7 + a + b = 0
Thus, to ensure that x – 1 divides our polynomial, a + b must be equal to -7.