Question

29 \( 6 x^{3}-48 x^{2}+5 x-40 \)

Ask by Stanley Martin. in the United States
Mar 12,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The polynomial \(6x^3 - 48x^2 + 5x - 40\) can be factored as \((x - 8)(6x^2 + 5)\).

Solution

1. Write the polynomial: \[ 6x^3 - 48x^2 + 5x - 40 \] 2. Group the terms: \[ \left(6x^3 - 48x^2\right) + \left(5x - 40\right) \] 3. Factor out the common factor from each group: - From the first group, factor out \(6x^2\): \[ 6x^3 - 48x^2 = 6x^2(x - 8) \] - From the second group, factor out \(5\): \[ 5x - 40 = 5(x - 8) \] 4. Notice the common factor \((x - 8)\) in both groups. Factor it out: \[ 6x^3-48x^2+5x-40 = (x-8)\left(6x^2+5\right) \] 5. The fully factored form of the polynomial is: \[ (x-8)(6x^2+5) \]

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The Deep Dive

Did you know that the factorization of polynomials dates back to ancient civilizations, including the Greeks? They used geometric methods to find areas and volumes, which essentially involved manipulating polynomial expressions long before the algebra we know today was formally developed. Mathematics really has a rich history steeped in exploration and discovery! In real-world applications, polynomials show up everywhere! Whether it's physics calculating trajectories, economics modeling growth, or computer graphics rendering, understanding how to work with polynomial expressions can unlock a treasure trove of practical solutions. So, next time you encounter a polynomial, think of all the exciting possibilities it represents beyond just the math homework!

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