Use long division to divide the polynomial 35x9−42x8−5x5+6x4 by 5x2−6x . Write your answer in standard form.(1 point)

Answer 1

To divide the polynomial \( 35x^9 - 42x^8 - 5x^5 + 6x^4 \) by \( 5x^2 - 6x \) using long division, we'll follow the steps systematically.

1. Start with the dividend and divisor: \[ \text{Dividend} = 35x^9 - 42x^8 - 5x^5 + 6x^4 \] \[ \text{Divisor} = 5x^2 - 6x \]

2. Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{35x^9}{5x^2} = 7x^7 \]

3. Multiply the entire divisor by \( 7x^7 \): \[ 7x^7(5x^2 - 6x) = 35x^9 - 42x^8 \]

4. Subtract this from the original dividend: \[ (35x^9 - 42x^8 - 5x^5 + 6x^4) - (35x^9 - 42x^8) = 0 - 5x^5 + 6x^4 \] So, we have: \[ -5x^5 + 6x^4 \]

5. Repeat the division. Now we have: \[ \text{New Dividend} = -5x^5 + 6x^4 \] Divide the leading term: \[ \frac{-5x^5}{5x^2} = -x^3 \]

6. Multiply the entire divisor by \(-x^3\): \[ -x^3(5x^2 - 6x) = -5x^5 + 6x^4 \]

7. Subtract this from the current dividend: \[ (-5x^5 + 6x^4) - (-5x^5 + 6x^4) = 0 \]

8. Now combine the results: The result from the long division is: \[ 7x^7 - x^3 \]

Thus, the final answer, written in standard form, is: \[ \boxed{7x^7 - x^3} \]