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Find the product of (2q9+3q7)(−6q2+9) .(1 point) Responses −12q18+27q7 negative 12 q superscript 18 baseline plus 27 q superscript 7 baseline −12q11−36q9+27q7 negative 12 q superscript 11 baseline minus 36 q superscript 9 baseline plus 27 q superscript 7 baseline −12q11+27q7 negative 12 q superscript 11 baseline plus 27 q superscript 7 baseline −12q11+36q9+27q7

1 answer

To find the product of \((2q^9 + 3q^7)(-6q^2 + 9)\), we can use the distributive property (also known as the FOIL method for binomials). We will multiply each term in the first polynomial by each term in the second polynomial.

  1. Multiply \(2q^9\) by \(-6q^2\): \[ 2q^9 \cdot -6q^2 = -12q^{9+2} = -12q^{11} \]

  2. Multiply \(2q^9\) by \(9\): \[ 2q^9 \cdot 9 = 18q^9 \]

  3. Multiply \(3q^7\) by \(-6q^2\): \[ 3q^7 \cdot -6q^2 = -18q^{7+2} = -18q^{9} \]

  4. Multiply \(3q^7\) by \(9\): \[ 3q^7 \cdot 9 = 27q^7 \]

Now we sum all these products together: \[ -12q^{11} + 18q^9 - 18q^9 + 27q^7 \]

Combine like terms:

  • The \(q^9\) terms: \[ 18q^9 - 18q^9 = 0 \]
  • The final expression becomes: \[ -12q^{11} + 27q^7 \]

Thus, the product of \((2q^9 + 3q^7)(-6q^2 + 9)\) is: \[ -12q^{11} + 27q^{7} \]

So the answer is: \[ \boxed{-12q^{11} + 27q^7} \]