Alright, let’s solve the problem step-by-step.
We are given the equation:
[tex]\[ x^6 - 27 y^9 = (x^2 - 3 y^3)(x^4 + 3 x^2 y^? + 9 y^6) \][/tex]
Our goal is to find the missing exponent [tex]\( ? \)[/tex] in the term [tex]\( 3 x^2 y^? \)[/tex].
To solve for the missing exponent, we can expand the right-hand side and compare it to the left-hand side.
1. Distribute [tex]\( x^2 \)[/tex] across the terms inside the second parenthesis:
[tex]\[
x^2 \cdot (x^4 + 3 x^2 y^? + 9 y^6) = x^6 + 3 x^4 y^? + 9 x^2 y^6
\][/tex]
2. Distribute [tex]\(-3 y^3 \)[/tex] across the terms inside the second parenthesis:
[tex]\[
-3 y^3 \cdot (x^4 + 3 x^2 y^? + 9 y^6) = -3 y^3 x^4 - 9 y^3 x^2 y^? - 27 y^9
\][/tex]
3. Combine all the terms from both distributions:
[tex]\[
x^6 + 3 x^4 y^? + 9 x^2 y^6 - 3 y^3 x^4 - 9 y^3 x^2 y^? - 27 y^9
\][/tex]
We want the resulting polynomial after expansion to match [tex]\( x^6 - 27 y^9 \)[/tex].
4. Match the terms with [tex]\( x^6 \)[/tex] and [tex]\( y^9 \)[/tex]:
- The [tex]\( x^6 \)[/tex] term and [tex]\(-27 y^9 \)[/tex] are already clear:
[tex]\[
x^6 - 27 y^9
\][/tex]
5. Comparing the powers of [tex]\( y \)[/tex] in both expressions:
- For the term [tex]\( 3 x^4 y^? \)[/tex] to cancel out with [tex]\(-3 y^3 x^4 \)[/tex]:
[tex]\[
3 x^4 y^? \text{ must match with } -3 y^3 x^4
\][/tex]
- Therefore, [tex]\( ? \)[/tex] must be 3.
Hence, the value of the missing exponent is [tex]\( \boxed{3} \)[/tex].
