To perform long division of the polynomial \(-30x^6 + 48x^5 + 50x - 80\) by \(5x - 8\), we will follow these steps:
- Divide the leading term of the dividend by the leading term of the divisor.
- Multiply the entire divisor by this result.
- Subtract this from the original polynomial (the dividend).
- Repeat the process with the new polynomial (the remainder) until the degree of the remainder is less than the degree of the divisor.
Let's start the division:
Step 1: Divide the leading terms
- The leading term of the dividend is \(-30x^6\).
- The leading term of the divisor is \(5x\).
\[ \frac{-30x^6}{5x} = -6x^5 \]
Step 2: Multiply the divisor by \(-6x^5\)
\[ (-6x^5)(5x - 8) = -30x^6 + 48x^5 \]
Step 3: Subtract from the original polynomial
Now we perform the subtraction: \[ (-30x^6 + 48x^5 + 50x - 80) - (-30x^6 + 48x^5) = 50x - 80 \]
Step 4: Repeat the process with the new polynomial \(50x - 80\)
Now we take \(50x - 80\) and divide it by \(5x - 8\).
Divide the leading terms:
\[ \frac{50x}{5x} = 10 \]
Multiply the divisor by \(10\):
\[ 10(5x - 8) = 50x - 80 \]
Subtract:
\[ (50x - 80) - (50x - 80) = 0 \]
Since we reached a remainder of \(0\), we stop here.
Final Result
The division shows that: \[ -30x^6 + 48x^5 + 50x - 80 = (5x - 8)(-6x^5 + 10) \]
Thus, the quotient of the division is: \[ \boxed{-6x^5 + 10} \]
And there is no remainder.