To find the polynomial that the expression x^4 + 5x^3 + 3x + 15 is divided by to get the quotient x^3 + 3, we can use polynomial long division.
1. Start with the equation:
x^4 + 5x^3 + 3x + 15 = (x^3 + 3) * P(x)
2. Here, P(x) is the polynomial we want to find.
3. We can rearrange the equation:
P(x) = (x^4 + 5x^3 + 3x + 15) / (x^3 + 3)
4. To perform the division, divide the leading terms:
x^4 / x^3 = x
5. Now multiply x by x^3 + 3:
x * (x^3 + 3) = x^4 + 3x
6. Subtract this from the original polynomial:
(x^4 + 5x^3 + 3x + 15) – (x^4 + 3x) = 5x^3 + 15 – 3x = 5x^3 + 15 – 3x
7. Now simplify:
5x^3 – 3x + 15
8. Divide the leading terms again:
5x^3 / x^3 = 5
9. Now multiply 5 by x^3 + 3:
5 * (x^3 + 3) = 5x^3 + 15
10. Subtract this from the current polynomial:
(5x^3 – 3x + 15) – (5x^3 + 15) = -3x
11. The remainder is -3x, which indicates that:
x^4 + 5x^3 + 3x + 15 = (x^3 + 3)(x + 5) – 3x
Thus, the polynomial we divided by is:
x^3 + 3
In conclusion, the polynomial that gives a quotient of x^3 + 3 when dividing x^4 + 5x^3 + 3x + 15 is x + 5.