Dividing Polynomials: A StepbyStep 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 14: 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 24. 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.