Multiply and simplify.

Question

left parenthesis 6 a minus 7 b right parenthesis left parenthesis 9 a squared minus 2 ab plus 7 b squared right parenthesis
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Part 1
The answer is
  
enter your response here.
​(Simplify your​ answer.)

Answer 1

To multiply the expression \((6a - 7b)(9a^2 - 2ab + 7b^2)\), we will use the distributive property (also known as the FOIL method for binomials):

\[ (6a - 7b)(9a^2 - 2ab + 7b^2) = 6a(9a^2 - 2ab + 7b^2) - 7b(9a^2 - 2ab + 7b^2) \]

Now, distribute \(6a\) to each term in the second polynomial:

1. \(6a \cdot 9a^2 = 54a^3\)
2. \(6a \cdot (-2ab) = -12a^2b\)
3. \(6a \cdot 7b^2 = 42ab^2\)

So, the first part is:

\[ 54a^3 - 12a^2b + 42ab^2 \]

Next, distribute \(-7b\) to each term in the second polynomial:

1. \(-7b \cdot 9a^2 = -63a^2b\)
2. \(-7b \cdot (-2ab) = 14ab^2\)
3. \(-7b \cdot 7b^2 = -49b^3\)

So, the second part is:

\[ -63a^2b + 14ab^2 - 49b^3 \]

Now, combine all the terms:

\[ 54a^3 + (-12a^2b - 63a^2b) + (42ab^2 + 14ab^2) - 49b^3 \]

Combine like terms:

1. The \(a^3\) term: \(54a^3\)
2. The \(a^2b\) terms: \(-12a^2b - 63a^2b = -75a^2b\)
3. The \(ab^2\) terms: \(42ab^2 + 14ab^2 = 56ab^2\)
4. The \(b^3\) term: \(-49b^3\)

Putting it all together, the simplified result is:

\[ 54a^3 - 75a^2b + 56ab^2 - 49b^3 \]

Thus, the final answer is:

\[ \boxed{54a^3 - 75a^2b + 56ab^2 - 49b^3} \]