Questions: Divide. (9 u^2 x^7-6 u^5 x^4-15 u^6 x^7) div(3 u^5 x^4) Simplify your answer as much

Divide. (9 u^2 x^7-6 u^5 x^4-15 u^6 x^7) div(3 u^5 x^4) Simplify your answer as much
Transcript text: Divide. \[ \left(9 u^{2} x^{7}-6 u^{5} x^{4}-15 u^{6} x^{7}\right) \div\left(3 u^{5} x^{4}\right) \] Simplify your answer as much
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Solution

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Solution Steps

Step 1: Divide the Leading Terms

Divide the leading term of the dividend \( -15 u^{6} x^{7} \) by the leading term of the divisor \( 3 u^{5} x^{4} \): \[ \frac{-15 u^{6} x^{7}}{3 u^{5} x^{4}} = -5 u x^{3} \]

Step 2: Calculate the Remainder

Multiply the entire divisor \( 3 u^{5} x^{4} \) by the result from Step 1, and subtract from the original dividend: \[ -15 u^{6} x^{7} - (3 u^{5} x^{4})(-5 u x^{3}) = -6 u^{5} x^{4} + 9 u^{2} x^{7} \]

Step 3: Repeat the Division Process

Now, divide the new leading term \( -6 u^{5} x^{4} \) by the leading term of the divisor \( 3 u^{5} x^{4} \): \[ \frac{-6 u^{5} x^{4}}{3 u^{5} x^{4}} = -2 \]

Step 4: Calculate the New Remainder

Multiply the entire divisor \( 3 u^{5} x^{4} \) by the result from Step 3, and subtract from the previous remainder: \[ -6 u^{5} x^{4} - (3 u^{5} x^{4})(-2) = 9 u^{2} x^{7} \]

Step 5: Compile the Quotient and Remainder

The quotient from the division process is: \[ -5 u x^{3} - 2 \] The final remainder is: \[ 9 u^{2} x^{7} \]

Step 6: Write the Complete Division Expression

The complete division expression can be written as: \[ \frac{9 u^{2} x^{7} - 6 u^{5} x^{4} - 15 u^{6} x^{7}}{3 u^{5} x^{4}} = -5 u x^{3} - 2 + \frac{9 u^{2} x^{7}}{3 u^{5} x^{4}} \] This simplifies to: \[ -5 u x^{3} - 2 + \frac{3 x^{3}}{u^{3}} \]

Final Answer

\(\boxed{-5 u x^{3} - 2 + \frac{3 x^{3}}{u^{3}}}\)

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