Construct a table to find the missing term in the product (−9m6+12m5)(m2+2m+1)=−9m8−6m7+?+12m5 .(1 point)

Responses

33m6
33 m superscript 6 baseline

−9m6
negative 9 m superscript 6 baseline

3m6
3 m superscript 6 baseline

15m6

1 answer

To find the missing term in the product \((-9m^6 + 12m^5)(m^2 + 2m + 1) = -9m^8 - 6m^7 + ? + 12m^5\), we can start by expanding the left-hand side of the equation.

Let's break it down:

  1. First Term: \(-9m^6 \cdot m^2\) \[ -9m^6 \cdot m^2 = -9m^8 \]

  2. Second Term: \(-9m^6 \cdot 2m\) \[ -9m^6 \cdot 2m = -18m^7 \]

  3. Third Term: \(-9m^6 \cdot 1\) \[ -9m^6 \cdot 1 = -9m^6 \]

  4. Fourth Term: \(12m^5 \cdot m^2\) \[ 12m^5 \cdot m^2 = 12m^7 \]

  5. Fifth Term: \(12m^5 \cdot 2m\) \[ 12m^5 \cdot 2m = 24m^6 \]

  6. Sixth Term: \(12m^5 \cdot 1\) \[ 12m^5 \cdot 1 = 12m^5 \]

Now, we can combine all these terms:

From the calculations:

  • The \(m^8\) coefficient: \( -9m^8 \)
  • The \(m^7\) coefficient: \( -18m^7 + 12m^7 = -6m^7 \)
  • The \(m^6\) coefficient: \( -9m^6 + 24m^6 = 15m^6 \)
  • The \(m^5\) coefficient: \( +12m^5 \)

Putting it all together:

\[ (-9m^8 - 6m^7 + 15m^6 + 12m^5) \]

Therefore, the missing term represented by \(?\) is \(15m^6\).

Thus, the correct answer is:

15m^6