To divide the polynomial px = x³ + 3x² + 5x + 3 by the polynomial gx = x² + 2, we can use polynomial long division.
1. **Set up the division**: Write the dividend (px) and the divisor (gx). We want to find out how many times gx fits into px.
2. **Divide the leading term**: Divide the leading term of px, which is x³, by the leading term of gx, which is x². This gives us x.
3. **Multiply and subtract**: Multiply gx by x:
x * (x² + 2) = x³ + 2x. Now subtract this from px:
(x³ + 3x² + 5x + 3) – (x³ + 2x) = 3x² + 3x + 3.
4. **Repeat the process**: Now, divide the leading term of the new polynomial 3x² by the leading term of gx x². This gives 3. Multiply gx by 3:
3 * (x² + 2) = 3x² + 6. Subtract this from 3x² + 3x + 3:
(3x² + 3x + 3) – (3x² + 6) = 3x – 3.
5. **Final step**: The degree of our remainder (3x – 3) is less than the degree of gx, so we stop here. Thus, the result can be expressed as:
Result:
x + 3 with a remainder of (3x – 3).
So we can write the final answer as:
px ÷ gx = x + 3 + (3x – 3)/(x² + 2)