Solving Polynomial Division: (x^5  13x^4  120x + 80) / (x + 3)
This article will guide you through the process of dividing the polynomial (x^5  13x^4  120x + 80) by (x + 3) using long division.
Understanding Long Division with Polynomials
Long division for polynomials works on the same principle as numerical long division. We aim to find a quotient polynomial that, when multiplied by the divisor, results in a product that matches the dividend as closely as possible.
Steps for Long Division:

Set up the division: Arrange the dividend and divisor in a long division format. Make sure the terms are in descending order of their exponents.
_____________ x + 3  x^5  13x^4  120x + 80

Divide the leading terms: Divide the leading term of the dividend (x^5) by the leading term of the divisor (x). This gives us x^4.
x^4 _____________ x + 3  x^5  13x^4  120x + 80

Multiply the quotient term by the divisor: Multiply the quotient term (x^4) by the entire divisor (x + 3). This gives us x^5 + 3x^4.
x^4 _____________ x + 3  x^5  13x^4  120x + 80 x^5 + 3x^4

Subtract: Subtract the product from the dividend. This gives us 16x^4.
x^4 _____________ x + 3  x^5  13x^4  120x + 80 x^5 + 3x^4  16x^4

Bring down the next term: Bring down the next term from the dividend (120x).
x^4 _____________ x + 3  x^5  13x^4  120x + 80 x^5 + 3x^4  16x^4  120x

Repeat the process: Repeat steps 25 until there are no more terms to bring down.
x^4  16x^3 + 48x^2  144x + 312 x + 3  x^5  13x^4  120x + 80 x^5 + 3x^4  16x^4  120x 16x^4  48x^3  48x^3  120x 48x^3 + 144x^2  144x^2  120x 144x^2  432x  312x + 80 312x + 936  856
Result
The final result is:
(x^5  13x^4  120x + 80) / (x + 3) = x^4  16x^3 + 48x^2  144x + 312  856/(x+3)
This can be expressed as:
x^4  16x^3 + 48x^2  144x + 312 with a remainder of 856/(x+3).
Conclusion
Polynomial long division is a fundamental technique in algebra. Understanding this process will allow you to simplify polynomial expressions and solve complex problems.