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Divide. If there is a remainder, include it as a simplified fraction.\newline(5t3+16t216t)÷(t+4)(5t^3 + 16t^2 - 16t) \div (t + 4)\newline______

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Q. Divide. If there is a remainder, include it as a simplified fraction.\newline(5t3+16t216t)÷(t+4)(5t^3 + 16t^2 - 16t) \div (t + 4)\newline______
  1. Set Up Division: First, we set up the division of the polynomial by the binomial in long division format.
  2. Find First Term Quotient: We divide the first term of the polynomial, 5t35t^3, by the first term of the binomial, tt, to get the first term of the quotient, which is 5t25t^2.\newlineCalculation: 5t3÷t=5t25t^3 \div t = 5t^2
  3. Subtract and Multiply: We multiply the entire binomial (t+4)(t + 4) by the term we just found, 5t25t^2, and subtract the result from the polynomial.\newlineCalculation: (t+4)(5t2)=5t3+20t2(t + 4)(5t^2) = 5t^3 + 20t^2\newlineSubtraction: (5t3+16t216t)(5t3+20t2)=4t216t(5t^3 + 16t^2 - 16t) - (5t^3 + 20t^2) = -4t^2 - 16t
  4. Find Next Term Quotient: We divide the new leading term, 4t2-4t^2, by the first term of the binomial, tt, to get the next term of the quotient, which is 4t-4t.\newlineCalculation: 4t2÷t=4t-4t^2 \div t = -4t
  5. Subtract and Multiply: We multiply the entire binomial (t+4)(t + 4) by the term we just found, 4t-4t, and subtract the result from the remaining polynomial terms.\newlineCalculation: (t+4)(4t)=4t216t(t + 4)(-4t) = -4t^2 - 16t\newlineSubtraction: (4t216t)(4t216t)=0(-4t^2 - 16t) - (-4t^2 - 16t) = 0
  6. Complete Division: Since the remainder is 00, we have completed the division process.

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