Question
c) \( \left(k^{2}+2 k+1\right)\left(k^{2}-2 k-1\right) \)
Ask by Martinez George. in South Africa
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( k^{4} - 4k^{2} - 4k - 1 \).
Solution
Calculate or simplify the expression \( (k^2+2k+1)(k^2-2k-1) \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(k^{2}+2k+1\right)\left(k^{2}-2k-1\right)\)
- step1: Apply the distributive property:
\(k^{2}\times k^{2}-k^{2}\times 2k-k^{2}\times 1+2k\times k^{2}-2k\times 2k-2k\times 1+1\times k^{2}-1\times 2k-1\times 1\)
- step2: Multiply the terms:
\(k^{4}-2k^{3}-k^{2}+2k^{3}-4k^{2}-2k+k^{2}-2k-1\)
- step3: Collect like terms:
\(k^{4}-4k^{2}-4k-1\)
The expansion of the given expression \( \left(k^{2}+2 k+1\right)\left(k^{2}-2 k-1\right) \) is \( k^{4}-4k^{2}-4k-1 \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
Did you know that the expression \( (k^2 + 2k + 1) \) can actually be simplified? It factors into \( (k+1)^2 \), which is the square of a binomial! So, when you multiply \( (k+1)^2 \) with \( (k^2 - 2k - 1) \), it becomes easier to work with. Now, let’s consider how the factors can interact in real-world scenarios, like in physics when dealing with projectile motion. If you were calculating the trajectory of an object, expressions like this could help you derive the path, especially when you need to determine maximum height or time of flight. These algebraic manipulations are foundational in the science of movement!
