To find the remainder when a polynomial is divided by a linear factor like x – c, we can use the Remainder Theorem. According to this theorem, the remainder of the polynomial f(x) when divided by x – c is simply f(c).
For the first polynomial, f(x) = ax³ + 3x² + 3, we need to calculate f(4):
- f(4) = a(4)³ + 3(4)² + 3
- f(4) = 64a + 48 + 3
- f(4) = 64a + 51
So, the remainder when ax³ + 3x² + 3 is divided by x – 4 is 64a + 51.
For the second polynomial, g(x) = 2x³ + 5x, we also calculate g(4):
- g(4) = 2(4)³ + 5(4)
- g(4) = 2(64) + 20
- g(4) = 128 + 20
- g(4) = 148
Therefore, the remainder when 2x³ + 5x is divided by x – 4 is 148.
In conclusion, the remainders are:
- For ax³ + 3x² + 3: 64a + 51
- For 2x³ + 5x: 148