$(2a^3 - 3)(5a^3 - 2) = 10 a^6 + ? + 6$
Algebraic Equation Simplification
The goal is to find the term represented by the question mark (?) in the equation \( (2a^3 - 3)(5a^3 - 2) = 10 a^6 + ? + 6 \). This requires expanding the product of the two binomials on the left side.
Expand Polynomials
We multiply the two binomials using the distributive property (often remembered by the FOIL acronym: First, Outer, Inner, Last):
Expression: \( (2a^3 - 3)(5a^3 - 2) \)
- First terms: \( 2a^3 \times 5a^3 = 10a^{3+3} = 10a^6 \)
- Outer terms: \( 2a^3 \times (-2) = -4a^3 \)
- Inner terms: \( -3 \times 5a^3 = -15a^3 \)
- Last terms: \( -3 \times (-2) = 6
Adding these results gives: \( 10a^6 - 4a^3 - 15a^3 + 6 \)
Combine Like Terms
Combine the terms containing \( a^3 \):
\( -4a^3 - 15a^3 = (-4 - 15)a^3 = -19a^3 \)
The fully expanded form is: \( 10a^6 - 19a^3 + 6 \)
Determine Missing Term
Comparing the expanded form \( 10a^6 - 19a^3 + 6 \) to the given equation \( 10 a^6 + ? + 6 \), we can see that the missing term is \( -19a^3 \).
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