Final answer:
To divide the given polynomial by the monomial, coefficients are divided, and exponents with like bases are subtracted. The term with x remains unchanged, and the final simplified result is − 3z²u + 4x⁷u² + 9.
Step-by-step explanation:
To divide the polynomial − 6z⁷u⁴ + 4x⁷u² + 18z⁵u³ by 2z⁵u³, we need to divide each term of the polynomial individually by the monomial. We do this by subtracting the exponents of like bases and dividing the coefficients in each term.
For the first term, − 6z⁷u⁴ / 2z⁵u³:
- Divide coefficients: − 6 / 2 = − 3
- Subtract exponents of z: 7 - 5 = 2, so we have z²
- Subtract exponents of u: 4 - 3 = 1, so we have u
For the second term, there is an x⁷ term that cannot be reduced by the monomial with z basis, thus it remains unchanged.
For the third term, 18z⁵u³ / 2z⁵u³:
- Divide coefficients: 18 / 2 = 9
- Subtract exponents of z: 5 - 5 = 0, so z's exponent drops out
- Subtract exponents of u: 3 - 3 = 0, so u's exponent drops out
Thus, the simplified expression is − 3z²u + 4x⁷u² + 9.