Split (4x¹⁰ − 5x⁹ − 20x⁴)/(4x²) into separate fractions and simplify.

Step-by-step solution

1. 1

   Split the single fraction into three. A sum in the numerator can always be divided term by term — this is the distributive law for division, $\tfrac{a+b+c}{d}=\tfrac ad+\tfrac bd+\tfrac cd$. (The same is not true for a sum in the denominator, which is the usual confusion.)

   $\frac{4x^{10}}{4x^{2}}-\frac{5x^{9}}{4x^{2}}-\frac{20x^{4}}{4x^{2}}$

2. 2

   Simplify the first term. Divide coefficients and subtract exponents:

   $\frac{4x^{10}}{4x^{2}}=1\cdot x^{10-2}=x^{8}$

3. 3

   Simplify the second term. Here $5$ does not divide evenly by $4$, so a fraction survives:

   $\frac{5x^{9}}{4x^{2}}=\frac54x^{7}$

4. 4

   Simplify the third term.

   $\frac{20x^{4}}{4x^{2}}=5x^{2}$

5. 5

   Reassemble with the original signs.

   $\boxed{x^{8}-\frac54x^{7}-5x^{2}}\qquad(x\neq 0)$

6. 6

   Check by multiplying back. $4x^{2}\left(x^{8}-\tfrac54x^{7}-5x^{2}\right)=4x^{10}-5x^{9}-20x^{4}$ ✓, which is the original numerator. A quick numerical check at $x=1$: the original is $\tfrac{4-5-20}{4}=-\tfrac{21}{4}=-5.25$, and the answer gives $1-1.25-5=-5.25$ ✓.