Divide. If the polynomial does not divide evenly, include the remainder as a fraction. \( \left(8 z^{3}+61 z^{2}+35 z\right) \div(z+7) \) Submit

To divide the polynomial \( 8z^{3} + 61z^{2} + 35z \) by \( z + 7 \), we can use polynomial long division. 1. Divide the first term of the dividend \( 8z^3 \) by the first term of the divisor \( z \) to get \( 8z^2 \). 2. Multiply \( 8z^2 \) by \( z + 7 \) to get \( 8z^3 + 56z^2 \). 3. Subtract this from the original polynomial: \[ (8z^3 + 61z^2 + 35z) - (8z^3 + 56z^2) = 5z^2 + 35z \] 4. Now, divide the first term of the new dividend \( 5z^2 \) by \( z \) to get \( 5z \). 5. Multiply \( 5z \) by \( z + 7 \) to get \( 5z^2 + 35z \). 6. Subtract this from the current polynomial: \[ (5z^2 + 35z) - (5z^2 + 35z) = 0 \] At this point, we have completed the division with no remainder. Therefore, we can conclude that: \[ \frac{8z^{3}+61z^{2}+35z}{z+7} = 8z^{2} + 5z \] So the result of the division is \( 8z^2 + 5z \) with a remainder of \( 0 \).