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

Question

Responses

3m6
3 m superscript 6 baseline

−9m6
negative 9 m superscript 6 baseline

15m6
15 m superscript 6 baseline

33m6

Answer 1

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 2

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 3

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 4

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 5

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 6

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 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:

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