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Multiply. Write your answer in simplest form.\newline3p3−p25p+1×4p+52\frac{3p^3 - p^2}{5p + 1} \times \frac{4p + 5}{2}

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Full solution

Q. Multiply. Write your answer in simplest form.\newline3p3−p25p+1×4p+52\frac{3p^3 - p^2}{5p + 1} \times \frac{4p + 5}{2}
  1. Identify Expressions: Identify the expressions to be multiplied.\newlineWe have two fractions to multiply: (3p3−p2)/(5p+1)(3p^3 - p^2)/(5p + 1) and (4p+5)/2(4p + 5)/2.
  2. Multiply Numerators and Denominators: Multiply the numerators and the denominators of the fractions.\newlineTo multiply fractions, we multiply the numerators together and the denominators together.\newline(3p3−p2)×(4p+5)(3p^3 - p^2) \times (4p + 5) is the numerator, and (5p+1)×2(5p + 1) \times 2 is the denominator.
  3. Perform Numerator Multiplication: Perform the multiplication of the numerators.\newlineWe distribute (3p3−p2)(3p^3 - p^2) across (4p+5)(4p + 5).\newline(3p3−p2)×(4p+5)=3p3×4p+3p3×5−p2×4p−p2×5(3p^3 - p^2) \times (4p + 5) = 3p^3 \times 4p + 3p^3 \times 5 - p^2 \times 4p - p^2 \times 5\newline=12p4+15p3−4p3−5p2= 12p^4 + 15p^3 - 4p^3 - 5p^2\newline=12p4+11p3−5p2= 12p^4 + 11p^3 - 5p^2
  4. Perform Denominator Multiplication: Perform the multiplication of the denominators.\newlineWe multiply (5p+1)(5p + 1) by 22.\newline(5p+1)×2=10p+2(5p + 1) \times 2 = 10p + 2
  5. Combine Results: Combine the results to form the new fraction. The new fraction is (12p4+11p3−5p2)(10p+2)(12p^4 + 11p^3 - 5p^2) \over (10p + 2).
  6. Simplify Fraction: Simplify the fraction if possible.\newlineWe look for common factors in the numerator and the denominator. In this case, there are no common factors, so the fraction is already in its simplest form.

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