Question
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Ask by Fowler Simpson. in the United States
Jan 21,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

All expressions can be factored as .

Solution

Let’s analyze the expressions provided step by step.
  1. First Expression:
    This expression can be factored out by taking as a common factor. We can focus on factoring the polynomial inside the parentheses.
  2. Second Expression:
    Here, we can also factor out and then focus on the polynomial .
  3. Third Expression:
    This expression is already factored, showing multiplied by two other factors.
  4. Fourth Expression:
    This is identical to the third expression.
Now, let’s focus on factoring the polynomial from the second expression. We can use substitution to make it easier. Let , then we have:
Next, we will factor this quadratic expression.
The factors of that add up to are and . Thus, we can write:
Substituting back , we get:
Now, substituting this back into the second expression:
This matches the third and fourth expressions.
Thus, we can conclude that all expressions are equivalent and can be factored as:
In summary, the factored form of all the expressions is:

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The Deep Dive

The polynomial gives us insight into the behavior of polynomial functions. It’s interesting to note how polynomials can be factored to reveal their roots and potential intercepts. By breaking down this polynomial, we also learn how the coefficients can affect the graph’s steepness and direction, making factorization a great tool for visualizing their characteristics!
In real-world applications, polynomials like this one can model a variety of phenomena, such as physics problems involving motion or economics related to profit functions. By determining the roots of the polynomial through factoring, we can predict when a business breaks even or when an object reaches a certain height. This shows the practical importance of understanding polynomial functions and their factorizations!

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