What is the remainder when x³ + 3x² + 13x + 78 is divided by x – 4?

To find the remainder when a polynomial is divided by a linear binomial, we can use the Remainder Theorem. This theorem states that the remainder of the division of a polynomial f(x) by x – c is equal to f(c).

In this case, our polynomial is:

f(x) = x³ + 3x² + 13x + 78

We need to divide by x – 4, which means we will evaluate the polynomial at x = 4.

Now let’s calculate:

f(4) = (4)³ + 3(4)² + 13(4) + 78
     = 64 + 48 + 52 + 78
     = 64 + 48 = 112
     = 112 + 52 = 164
     = 164 + 78 = 242

So the remainder when x³ + 3x² + 13x + 78 is divided by x – 4 is 242.

In conclusion, the remainder is 242.