-6t^2(2t - t^4) = -12t^3 + 6t^6. - Decision Point
Simplifying and Verifying the Expression: A Comprehensive Guide to –6t²(2t – t⁴) = –12t³ + 6t⁶
Simplifying and Verifying the Expression: A Comprehensive Guide to –6t²(2t – t⁴) = –12t³ + 6t⁶
When reducing algebraic expressions or verifying equation equivalences, clear step-by-step manipulation is key. One such expression—–6t²(2t – t⁴)—often appears in algebra students’ practice, and understanding its expansion and simplification is essential for mastering polynomial manipulation. This article breaks down the equation –6t²(2t – t⁴) = –12t³ + 6t⁶ thoroughly, showing how to simplify the left-hand side and verify its equivalence with the right-hand side.
Understanding the Context
What Is the Equation?
We aim to confirm whether:
–6t²(2t – t⁴) = –12t³ + 6t⁶
This involves multiplying the binomial inside the parentheses by the polynomial outside and collecting like terms, then comparing with the given right-hand side.
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Key Insights
Step 1: Distribute the Polynomial
Start by distributing –6t² across each term inside the parentheses:
–6t²(2t – t⁴) = (–6t²)(2t) + (–6t²)(–t⁴)
Now calculate each product:
- (–6t²)(2t) = –12t³
- (–6t²)(–t⁴) = +6t⁶ (because a negative times a negative is positive)
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Step 2: Combine Like Terms
Adding the two results:
–12t³ + 6t⁶
So, the left-hand side simplifies to:
–6t²(2t – t⁴) = –12t³ + 6t⁶
Step 3: Verify Equivalence
Now observe that:
Left-hand side (LHS): –6t²(2t – t⁴) → after expansion: –12t³ + 6t⁶
Right-hand side (RHS): –12t³ + 6t⁶
Both sides are identical. Therefore, the original equation holds true.
