Question
1. \( 8 m^{12}+1 \)
Ask by Lynch Frazier. in South Africa
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 8 m^{12} + 1 \) can be factored into \( (2m^{4} + 1)(4m^{8} - 2m^{4} + 1) \).
Solution
Factor the expression by following steps:
- step0: Factor:
\(8m^{12}+1\)
- step1: Rewrite the expression:
\(\left(2m^{4}\right)^{3}+1^{3}\)
- step2: Factor the expression:
\(\left(2m^{4}+1\right)\left(\left(2m^{4}\right)^{2}-2m^{4}\times 1+1^{2}\right)\)
- step3: Evaluate:
\(\left(2m^{4}+1\right)\left(4m^{8}-2m^{4}+1\right)\)
The expression \( 8 m^{12} + 1 \) can be factored as follows:
\[
8 m^{12} + 1 = (2m^{4} + 1)(4m^{8} - 2m^{4} + 1)
\]
This shows the factorization into two polynomials.
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Simplify this solution Bonus Knowledge
Let's have some fun with \( 8 m^{12}+1 \)! This expression can be factored using the sum of cubes formula. Remember that \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)? Here, we can rewrite \( 8 m^{12} \) as \( (2 m^4)^3 \) and \( 1 \) as \( 1^3 \). So we can express it as \( (2 m^4 + 1)((2 m^4)^2 - (2 m^4)(1) + 1^2) \). This simplifies beautifully! Now, think about how this factoring might pop up in real life! Polynomial identities like this help in coding algorithms, shaping curves in computer graphics, or even optimizing equations in engineering. They give you the tools to simplify complex systems and make calculations easier, whether you’re designing software or analyzing data!
