(x^5-13x^4-120x+80)/(x+3)

4 min read Jun 17, 2024

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 2-5 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.