To solve the division of the polynomial 2x3 + 9x2 + 11x + 6 by x3, we can apply polynomial long division or simply simplify the expression, since dividing by x3 means reducing the polynomial by that degree.
First, let’s express the division:
Division: rac{2x^3 + 9x^2 + 11x + 6}{x^3}
Next, we can break it down term by term:
- For the first term: rac{2x^3}{x^3} = 2
- For the second term: rac{9x^2}{x^3} = rac{9}{x}
- For the third term: rac{11x}{x^3} = rac{11}{x^2}
- For the constant term: rac{6}{x^3}
So now we can put it all together:
Result: The result of the division is:
2 + rac{9}{x} + rac{11}{x^2} + rac{6}{x^3}
This expression shows that the polynomial is reduced based on the degree of the polynomial after the division. As x approaches infinity, the terms involving x in the denominator will approach zero, leaving the constant term as the dominating factor. Therefore, for large values of x, the result approaches 2.
In summary, dividing 2x3 + 9x2 + 11x + 6 by x3 yields 2 + rac{9}{x} + rac{11}{x^2} + rac{6}{x^3}.