Final answer:
To divide the polynomials f(x) by g(x), apply polynomial long division by dividing each term systematically, eliminating terms to simplify, and checking the result for reasonableness.
Step-by-step explanation:
Dividing Polynomials Step by Step
To divide the given polynomials f(x) by g(x), we can use long division or synthetic division (since the divisor is a second degree polynomial). We'll proceed with polynomial long division.
Set up the division, writing f(x) under the long division symbol and g(x) outside.
Divide the first term of f(x), which is 3x4, by the first term of g(x), which is x2, to get 3x2. Write this above the division symbol.
Multiply 3x2 by g(x) and subtract this from f(x), then bring down the next term of the dividend.
Repeat this process until all the terms have been brought down and the remainder is either zero or of lesser degree than the divisor.
Eliminate terms wherever possible to simplify the algebra.
Check the answer to see if it is reasonable, verifying there's no remainder of a degree equal to or higher than the divisor.
Example:
3x2 as the first term of the quotient means you would perform 3x2 * (x2 + 3x - 1) and subtract this from f(x) to continue with the division.
By following these steps methodically, the division should result in a smaller degree polynomial plus a remainder, which represents the division of f(x) over g(x).