(x^4+10x^3+8x^2-59x+40)/(x^2+3x-5)

5 min read Jun 17, 2024

Dividing Polynomials: A Step-by-Step Guide

This article will guide you through the process of dividing the polynomial x⁴ + 10x³ + 8x² - 59x + 40 by x² + 3x - 5. We will utilize polynomial long division, a method analogous to long division with numbers.

Setting up the Division

Arrange the terms: Write the dividend (x⁴ + 10x³ + 8x² - 59x + 40) and divisor (x² + 3x - 5) in descending order of their exponents.
Create the division format: Arrange the polynomials in a long division format.
```
    ___________
x² + 3x - 5 | x⁴ + 10x³ + 8x² - 59x + 40 
```

Performing the Division

Divide the leading terms: Divide the leading term of the dividend (x⁴) by the leading term of the divisor (x²). This gives us x². Write this quotient above the x³ term in the quotient area.
```
    x²        
x² + 3x - 5 | x⁴ + 10x³ + 8x² - 59x + 40 
```
Multiply the quotient term by the divisor: Multiply the quotient term (x²) by the entire divisor (x² + 3x - 5). This gives us x⁴ + 3x³ - 5x². Write this result below the dividend.
```
    x²        
x² + 3x - 5 | x⁴ + 10x³ + 8x² - 59x + 40 
               x⁴ + 3x³ - 5x² 
```

Subtract: Subtract the result from the dividend. Remember to change the signs of the terms being subtracted.

    x²        
x² + 3x - 5 | x⁴ + 10x³ + 8x² - 59x + 40 
               x⁴ + 3x³ - 5x² 
               ---------
                    7x³ + 13x² - 59x

Bring down the next term: Bring down the next term of the dividend (-59x).

    x²        
x² + 3x - 5 | x⁴ + 10x³ + 8x² - 59x + 40 
               x⁴ + 3x³ - 5x² 
               ---------
                    7x³ + 13x² - 59x

Repeat steps 1-4: Repeat the process of dividing the leading term of the new dividend (7x³) by the leading term of the divisor (x²). This gives us 7x. Write this quotient next to the previous one.

    x² + 7x   
x² + 3x - 5 | x⁴ + 10x³ + 8x² - 59x + 40 
               x⁴ + 3x³ - 5x² 
               ---------
                    7x³ + 13x² - 59x 
                    7x³ + 21x² - 35x

Continue the process: Repeat steps 2-4. Multiply the quotient term (7x) by the divisor, subtract, and bring down the next term (40).

    x² + 7x   
x² + 3x - 5 | x⁴ + 10x³ + 8x² - 59x + 40 
               x⁴ + 3x³ - 5x² 
               ---------
                    7x³ + 13x² - 59x 
                    7x³ + 21x² - 35x
                    ---------
                         -8x² - 24x + 40

Final step: Repeat the division process one more time. Divide the leading term of the new dividend (-8x²) by the leading term of the divisor (x²). This gives us -8. Write this quotient next to the previous one.

    x² + 7x - 8
x² + 3x - 5 | x⁴ + 10x³ + 8x² - 59x + 40 
               x⁴ + 3x³ - 5x² 
               ---------
                    7x³ + 13x² - 59x 
                    7x³ + 21x² - 35x
                    ---------
                         -8x² - 24x + 40
                         -8x² - 24x + 40 
                         ---------
                                 0

The Result

We have reached a remainder of 0. Therefore, the polynomial x⁴ + 10x³ + 8x² - 59x + 40 divided by x² + 3x - 5 is x² + 7x - 8.

Conclusion

Polynomial long division is a useful tool for dividing polynomials. It is essential to be careful with the signs when subtracting and to arrange the terms in descending order of their exponents.