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Find the product of (4x10+9x7)(−4x10−9x7) .(1 point) Responses −16x20−72x17−81x14 negative 16 x to the 20th power minus 72 x to the 17th power minus 81 x to the 14th power −16x20−81x14 negative 16 x to the 20th power minus 81 x to the 14th power 16x20−72x17+81x14 16 x to the 20th power minus 72 x to the 17th power plus 81 x to the 14th power −16x100−72x70−81x49

1 answer

To find the product of \((4x^{10} + 9x^7)(-4x^{10} - 9x^7)\), we can use the distributive property (also known as the FOIL method for binomials).

Let's denote:

  • \(A = 4x^{10}\)
  • \(B = 9x^7\)

Now, we rewrite the expression as: \[ (A + B)(-A - B) \]

Now we can expand this:

\[ = A(-A) + A(-B) + B(-A) + B(-B) \] \[ = -A^2 - AB - AB - B^2 \] \[ = -A^2 - 2AB - B^2 \]

Calculating each term:

  1. Calculate \(A^2 = (4x^{10})^2 = 16x^{20}\).
  2. Calculate \(B^2 = (9x^7)^2 = 81x^{14}\).
  3. Calculate \(AB = 4x^{10} \cdot 9x^7 = 36x^{17}\).

Substituting these values into the expanded expression:

\[ = -16x^{20} - 2(36x^{17}) - 81x^{14} \] \[ = -16x^{20} - 72x^{17} - 81x^{14} \]

Thus, the product is: \[ -16x^{20} - 72x^{17} - 81x^{14} \]

The correct response is: \(-16x^{20} - 72x^{17} - 81x^{14}\)