Questions: Divide and check (45 c^8 p^9-40 c^6 p^7+20 c^3 p^4)/(5 c^3 p)

Divide and check (45 c^8 p^9-40 c^6 p^7+20 c^3 p^4)/(5 c^3 p)
Transcript text: Divide and check \[ \frac{45 c^{8} p^{9}-40 c^{6} p^{7}+20 c^{3} p^{4}}{5 c^{3} p} \]
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Solution

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Solution Steps

To solve the given expression, we need to divide each term in the numerator by the denominator. This involves simplifying the coefficients and reducing the powers of the variables.

Step 1: Simplify the Expression

To simplify the given expression: \[ \frac{45 c^{8} p^{9} - 40 c^{6} p^{7} + 20 c^{3} p^{4}}{5 c^{3} p} \]

Step 2: Divide Each Term by the Denominator

Divide each term in the numerator by the denominator \(5 c^{3} p\): \[ \frac{45 c^{8} p^{9}}{5 c^{3} p} - \frac{40 c^{6} p^{7}}{5 c^{3} p} + \frac{20 c^{3} p^{4}}{5 c^{3} p} \]

Step 3: Simplify Each Term

Simplify each term individually: \[ \frac{45 c^{8} p^{9}}{5 c^{3} p} = 9 c^{5} p^{8} \] \[ \frac{40 c^{6} p^{7}}{5 c^{3} p} = 8 c^{3} p^{6} \] \[ \frac{20 c^{3} p^{4}}{5 c^{3} p} = 4 p^{3} \]

Step 4: Combine the Simplified Terms

Combine the simplified terms to get the final expression: \[ 9 c^{5} p^{8} - 8 c^{3} p^{6} + 4 p^{3} \]

Final Answer

\[ \boxed{p^{3} (9 c^{5} p^{5} - 8 c^{3} p^{3} + 4)} \]

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