-6t^2(2t - t^4) = -12t^3 + 6t^6.
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.
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.
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)
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.
Why This Matters: The Importance of Polynomial Expansion
Expanding expressions like this is a fundamental skill in algebra with broad applications:
- Solving equations: Simplifying both sides helps isolate variables.
- Graphing functions: Understanding polynomial structure reveals roots and end behavior.
- Calculus applications: Expanded forms support differentiation and integration.
Mastering techniques such as distributing binomials and combining like terms builds confidence and accuracy in algebra.
Summary
- The expression –6t²(2t – t⁴) expands to –12t³ + 6t⁶.
- This confirms the equivalence: –6t²(2t – t⁴) = –12t³ + 6t⁶ is true.
- Key steps include distribution, handling signs (especially negative × negative), and simplifying terms.
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Final Notes
Remember: Polynomial expressions are tools for modeling relationships in math, science, and engineering. Mastery of expansion, factoring, and simplification ensures you’re prepared for advanced topics like calculus, linear algebra, and differential equations. Practice transforms complexity into clarity—keep practicing!
For more algebra help, explore our tutorials on binomial expansion, combining like terms, and equation solving strategies.
Key takeaway: Expanding –6t²(2t – t⁴) step-by-step yields –12t³ + 6t⁶, confirming the equation’s validity and reinforcing essential algebra skills.
