Question

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

3m6
3 m superscript 6 baseline

−9m6
negative 9 m superscript 6 baseline

15m6
15 m superscript 6 baseline

33m6

Answer 1

Multiply the polynomials (6s4+15s3)(6s3−15s4) by constructing a table.(1 point)
Responses

−90s8−189s7+90s6
negative 90 s superscript 8 baseline minus 189 s superscript 7 baseline plus 90 s superscript 6 baseline

36s8−225s6
36 s superscript 8 baseline minus 225 s superscript 6 baseline

−198s7
negative 198 s superscript 7 baseline

90s8−189s7+90s6

Answer 2

Use the Distributive Property to multiply the polynomials −5t3(6t7−9t) .(1 point)
Responses

30t10+45t4
30 t superscript 10 baseline plus 45 t superscript 4 baseline

−30t21+45t3
negative 30 t superscript 21 baseline plus 45 t cubed

−30t10−14t4
negative 30 t superscript 10 baseline minus 14 t superscript 4 baseline

−30t10+45t4

Answer 3

Consider the product (2x−x3)(−3x4−7x2) . Which of the following is the correct expression when each term of the first polynomial is distributed onto the second polynomial?(1 point)
Responses

2x(−3x4−7x2)+x3(−3x4−7x2)
2 x left parenthesis negative 3 x superscript 4 baseline minus 7 x squared right parenthesis plus x cubed left parenthesis negative 3 x superscript 4 baseline minus 7 x squared right parenthesis

2x(−3x4−7x2)−x3(−3x4−7x2)
2 x left parenthesis negative 3 x superscript 4 baseline minus 7 x squared right parenthesis minus x cubed left parenthesis negative 3 x superscript 4 baseline minus 7 x squared right parenthesis

2x(−x3−3x4−7x2)
2 x left parenthesis negative 3 x cubed minus 3 x superscript 4 baseline minus 7 x squared right parenthesis

2x(−3x4)−x3(−7x2)

Answer 4

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

### Expansion Steps:

1. Distributing \(-9m^6\) with each term in \((m^2 + 2m + 1)\):
- \(-9m^6 \cdot m^2 = -9m^8\)
- \(-9m^6 \cdot 2m = -18m^7\)
- \(-9m^6 \cdot 1 = -9m^6\)

2. Distributing \(12m^5\) with each term in \((m^2 + 2m + 1)\):
- \(12m^5 \cdot m^2 = 12m^7\)
- \(12m^5 \cdot 2m = 24m^6\)
- \(12m^5 \cdot 1 = 12m^5\)

### Combine Like Terms:

Now, combine all the terms obtained from the distribution:

- The \(m^8\) term:
- \(-9m^8\)

- The \(m^7\) terms:
- \(-18m^7 + 12m^7 = -6m^7\)

- The \(m^6\) terms:
- \(-9m^6 + 24m^6 = 15m^6\)

- The \(m^5\) term:
- \(12m^5\)

### Result:
So, the full expanded product is:
\[
-9m^8 - 6m^7 + 15m^6 + 12m^5
\]

From the expression given in the problem:
\[
-9m^8 - 6m^7 + ? + 12m^5
\]
we can now see that \(?\) corresponds to \(15m^6\).

### Conclusion:
The missing term is:

\[
\boxed{15m^6}
\]

Answer 5

To multiply the polynomials \((6s^4 + 15s^3)(6s^3 - 15s^4)\), we can construct a table and calculate the products of each pair of terms.

### Step-by-step Table Construction:

1. **List the terms of each polynomial**:
- From the first polynomial: \(6s^4\), \(15s^3\)
- From the second polynomial: \(6s^3\), \(-15s^4\)

2. **Construct the table**:

| | \(6s^3\) | \(-15s^4\) |
|--------|-------------------|-------------------|
| \(6s^4\) | \(6s^4 \cdot 6s^3 = 36s^7\) | \(6s^4 \cdot -15s^4 = -90s^8\) |
| \(15s^3\)| \(15s^3 \cdot 6s^3 = 90s^6\) | \(15s^3 \cdot -15s^4 = -225s^7\) |

### Step 3: Combine the results:

Now, we will combine the results from the table:

- **From the first row**:
- \(36s^7\) from \(6s^4 \cdot 6s^3\)
- \(-90s^8\) from \(6s^4 \cdot -15s^4\)

- **From the second row**:
- \(90s^6\) from \(15s^3 \cdot 6s^3\)
- \(-225s^7\) from \(15s^3 \cdot -15s^4\)

### Combine like terms:
- The \(s^8\) term:
- \(-90s^8\)

- The \(s^7\) terms:
- \(36s^7 - 225s^7 = -189s^7\)

- The \(s^6\) term:
- \(90s^6\)

### Final Result:
So, the final result of the multiplication is:
\[
-90s^8 - 189s^7 + 90s^6
\]

### Conclusion:
The correct response is:

\[
\boxed{-90s^8 - 189s^7 + 90s^6}
\]

Answer 6

To multiply the polynomial \(-5t^3(6t^7 - 9t)\) using the Distributive Property, we will distribute \(-5t^3\) to each term inside the parentheses.

### Step-by-step Distribution:

1. **First term**:
\[
-5t^3 \cdot 6t^7 = -30t^{3+7} = -30t^{10}
\]

2. **Second term**:
\[
-5t^3 \cdot -9t = 45t^{3+1} = 45t^{4}
\]

### Combine the results:

Now we combine the results from both distributions:

\[
-30t^{10} + 45t^{4}
\]

### Conclusion:

Thus, the result of multiplying the polynomials is:

\[
\boxed{-30t^{10} + 45t^{4}}
\]

Answer 7

To distribute each term of the first polynomial \((2x - x^3)\) onto the second polynomial \((-3x^4 - 7x^2)\), we should multiply both terms of the first polynomial by each term of the second polynomial.

### Distributing the Terms:

1. **First term \(2x\)** distributed to \(-3x^4\) and \(-7x^2\):
- \(2x \cdot (-3x^4) = -6x^5\)
- \(2x \cdot (-7x^2) = -14x^3\)

2. **Second term \(-x^3\)** distributed to \(-3x^4\) and \(-7x^2\):
- \(-x^3 \cdot (-3x^4) = 3x^7\)
- \(-x^3 \cdot (-7x^2) = 7x^5\)

### Forming the Correct Expression:

Putting these all together, we can express the distribution as:

\[
2x(-3x^4 - 7x^2) - x^3(-3x^4 - 7x^2)
\]

### Conclusion:

The correct expression when each term of the first polynomial is distributed onto the second polynomial is:

\[
\boxed{2x(-3x^4 - 7x^2) - x^3(-3x^4 - 7x^2)}
\]