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Divide. If there is a remainder, include it as a simplified fraction.\newline(4w3+13w2+3w)÷(w+3)(4w^3 + 13w^2 + 3w) \div (w + 3)\newline______

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Q. Divide. If there is a remainder, include it as a simplified fraction.\newline(4w3+13w2+3w)÷(w+3)(4w^3 + 13w^2 + 3w) \div (w + 3)\newline______
  1. Use Polynomial Long Division: To divide the polynomial (4w3+13w2+3w)(4w^3 + 13w^2 + 3w) by the binomial (w+3)(w + 3), we will use polynomial long division.\newlineFirst, we divide the leading term of the polynomial, 4w34w^3, by the leading term of the binomial, ww, to get the first term of the quotient.\newline4w3÷w=4w24w^3 \div w = 4w^2\newlineNow, we multiply the entire binomial (w+3)(w + 3) by this term (4w2)(4w^2) and subtract the result from the original polynomial.\newline(4w3+13w2+3w)(4w2(w+3))=(4w3+13w2+3w)(4w3+12w2)(4w^3 + 13w^2 + 3w) - (4w^2 \cdot (w + 3)) = (4w^3 + 13w^2 + 3w) - (4w^3 + 12w^2)
  2. Find First Quotient Term: After subtracting, we combine like terms to find the new polynomial to divide.\newline(4w3+13w2+3w)(4w3+12w2)=w2+3w(4w^3 + 13w^2 + 3w) - (4w^3 + 12w^2) = w^2 + 3w\newlineNow, we divide the leading term of the new polynomial, w2w^2, by the leading term of the binomial, ww, to get the next term of the quotient.\newlinew2÷w=ww^2 \div w = w\newlineWe multiply the entire binomial (w+3)(w + 3) by this term (w)(w) and subtract the result from the new polynomial.\newline(w2+3w)(w(w+3))=(w2+3w)(w2+3w)(w^2 + 3w) - (w \cdot (w + 3)) = (w^2 + 3w) - (w^2 + 3w)
  3. Subtract and Combine Like Terms: After subtracting, we find that the result is 00, which means there is no remainder.(w2+3w)(w2+3w)=0(w^2 + 3w) - (w^2 + 3w) = 0Therefore, the division is exact, and there is no remainder to express as a fraction. The quotient we have found is 4w2+w4w^2 + w.

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