To divide the polynomial \(35x^9−42x^8−5x^5+6x^4\) by \(5x^2−6x\) using long division, we will proceed through the standard long division steps.
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Set up the division.
We want to divide \(35x^9 - 42x^8 - 5x^5 + 6x^4\) (the dividend) by \(5x^2 - 6x\) (the divisor).
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Divide the leading term.
Divide the leading term of the dividend \(35x^9\) by the leading term of the divisor \(5x^2\):
\[ \frac{35x^9}{5x^2} = 7x^7 \]
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Multiply and subtract.
Multiply the entire divisor \(5x^2 - 6x\) by \(7x^7\):
\[ 7x^7 \cdot (5x^2 - 6x) = 35x^9 - 42x^8 \]
Now subtract this from the original polynomial:
\[ (35x^9 - 42x^8 - 5x^5 + 6x^4) - (35x^9 - 42x^8) = -5x^5 + 6x^4 \]
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Repeat the process.
Now divide the leading term \(-5x^5\) by \(5x^2\):
\[ \frac{-5x^5}{5x^2} = -x^3 \]
Multiply the divisor \(5x^2 - 6x\) by \(-x^3\):
\[ -x^3 \cdot (5x^2 - 6x) = -5x^5 + 6x^4 \]
Subtract this from \(-5x^5 + 6x^4\):
\[ (-5x^5 + 6x^4) - (-5x^5 + 6x^4) = 0 \]
Since we reached 0, the division has completed.
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Result.
The quotient from the long division is:
\[ 7x^7 - x^3 \]
Therefore, the final answer in standard form is:
\[ \boxed{7x^7 - x^3} \]