To find the remainders when dividing the polynomials ax³ + 3x² + 3 and 2x³ + 5x by x – 4, we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial f(x) by x – c is simply f(c).
Step 1: Calculate for the first polynomial
Let f(x) = ax³ + 3x² + 3. We need to evaluate f(4):
- f(4) = a(4)³ + 3(4)² + 3
- = a(64) + 3(16) + 3
- = 64a + 48 + 3
- = 64a + 51
Thus, when ax³ + 3x² + 3 is divided by x – 4, the remainder is 64a + 51.
Step 2: Calculate for the second polynomial
Now consider the second polynomial, g(x) = 2x³ + 5x. We will evaluate g(4):
- g(4) = 2(4)³ + 5(4)
- = 2(64) + 5(4)
- = 128 + 20
- = 148
Therefore, when 2x³ + 5x is divided by x – 4, the remainder is 148.
Conclusion
In summary, the remainders of the given polynomials when divided by x – 4 are:
- For ax³ + 3x² + 3: 64a + 51
- For 2x³ + 5x: 148