Question

Find the product of (4x^10+9x^7)(−4x^10−9x^7) .

Answer 1

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

Let's denote:

- \(A = 4x^{10}\)
- \(B = 9x^{7}\)

So, we can rewrite the expression as:

\[(A + B)(-A - B)\]

Now, applying the distributive property:

\[
= A \cdot (-A) + A \cdot (-B) + B \cdot (-A) + B \cdot (-B)
\]

Calculating each term:

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

Now, combine the terms:

\[
-16x^{20} - 36x^{17} - 36x^{17} - 81x^{14}
\]

Combine the like terms \((-36x^{17} - 36x^{17} = -72x^{17})\):

\[
-16x^{20} - 72x^{17} - 81x^{14}
\]

Thus, the final answer is:

\[
\boxed{-16x^{20} - 72x^{17} - 81x^{14}}
\]