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

Question

Responses

15m6
15 m superscript 6 baseline

3m6
3 m superscript 6 baseline

−9m6
negative 9 m superscript 6 baseline

33m6

Answer 1

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

We can use the distributive property (also referred to as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.

1. Distributing \(-9m^6\):

   • \(-9m^6 \cdot m^2 = -9m^8\)
   • \(-9m^6 \cdot 2m = -18m^7\)
   • \(-9m^6 \cdot 1 = -9m^6\)

2. Distributing \(12m^5\):

   • \(12m^5 \cdot m^2 = 12m^7\)
   • \(12m^5 \cdot 2m = 24m^6\)
   • \(12m^5 \cdot 1 = 12m^5\)

Now, we can combine the terms we calculated:

\[ -9m^8 + (-18m^7 + 12m^7) + (-9m^6 + 24m^6) + 12m^5 \]

This simplifies to:

1. \(-9m^8\)
2. \((-18m^7 + 12m^7) = -6m^7\)
3. \((-9m^6 + 24m^6) = 15m^6\)
4. \(12m^5\)

Putting all of this together, we get:

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

Now we can see that the missing term marked by the question mark is \(15m^6\).

Therefore, the correct response is:

15m⁶