Simplify (x⁵ − x + 20x³ − 8) divided by (7 + x² − 2x).

Step-by-step solution

1. 1

   Put both polynomials in standard descending order. As written the terms are scrambled, and long division depends entirely on degree order:

   $\text{numerator: } x^5 + 20x^3 - x - 8, \qquad \text{denominator: } x^2 - 2x + 7$

   The quotient will have degree $5 - 2 = 3$.

2. 2

   Insert zero placeholders for the missing powers. The numerator has no $x^4$ and no $x^2$ term, so write

   $x^5 + 0x^4 + 20x^3 + 0x^2 - x - 8$

   Without these placeholders the columns misalign and every subsequent subtraction goes wrong.

3. 3

   Rounds 1 and 2. Divide leading terms, multiply, subtract:

   $\frac{x^5}{x^2} = x^3: \quad x^3(x^2-2x+7) = x^5 - 2x^4 + 7x^3 \ \Rightarrow\ 2x^4 + 13x^3 + 0x^2$

   $\frac{2x^4}{x^2} = 2x^2: \quad 2x^2(x^2-2x+7) = 2x^4 - 4x^3 + 14x^2 \ \Rightarrow\ 17x^3 - 14x^2 - x$

4. 4

   Rounds 3 and 4.

   $\frac{17x^3}{x^2} = 17x: \quad 17x(x^2-2x+7) = 17x^3 - 34x^2 + 119x \ \Rightarrow\ 20x^2 - 120x - 8$

   $\frac{20x^2}{x^2} = 20: \quad 20(x^2-2x+7) = 20x^2 - 40x + 140 \ \Rightarrow\ -80x - 148$

   The leftover $-80x - 148$ has degree $1$, below the divisor degree $2$, so the division stops here.

5. 5

   Write the result as quotient plus proper remainder.

   $\frac{x^5 + 20x^3 - x - 8}{x^2 - 2x + 7} = x^3 + 2x^2 + 17x + 20 - \frac{80x + 148}{x^2 - 2x + 7}$

6. 6

   Verify by multiplying back. Expanding $(x^2 - 2x + 7)(x^3 + 2x^2 + 17x + 20) + (-80x - 148)$ reproduces $x^5 + 20x^3 - x - 8$ exactly. Note the divisor $x^2 - 2x + 7$ has discriminant $4 - 28 = -24 < 0$, so it never factors over the reals and the remainder cannot be simplified further by partial fractions with real linear factors.