The result of the division is 2x4 - 2/3x²+2/3x with no remainder.
To solve the given expression (4x^3 + 7x^5 + 9x + 3x^2)/(3x + 1), you'll typically use polynomial long division or synthetic division, depending on the degree of the polynomials involved. In this case, I'll use polynomial long division.
Make sure that the terms are arranged in descending order of the variable's exponent:
7x5+423 + 3x²+9x
3x + 1|4x3+0x2 + 9x +0
Now, you want to divide the highest degree term of the numerator by the highest degree term of the denominator. In this case, it's 725/32. The result is 2x4. 2x4. Multiply the entire denominator by 224 and subtract it from the numerator:
7x5+423 + 3x²+9x 2
-(6x²+2x3)
025+2x3+3x²+9x
Now, repeat the process. Divide the highest degree term of the new numerator by the highest degree term of the denominator: 223/32. The result is 2/3x². Multiply the entire denominator by 2/3x² and subtract it from the numerator:
2x3+3x²+9x -(2/3x²+2/3x)
0x3+7/3x²+25/3x
Repeat the process again:
7/3x²+25/3x
(3x+1)
0
So, the result of the division is 2x4 - 2/3x²+2/3x with no remainder.