Question
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Answer
All expressions can be factored as
.
Solution
Let’s analyze the expressions provided step by step.
-
First Expression:This expression can be factored out by taking
as a common factor. We can focus on factoring the polynomial inside the parentheses. -
Second Expression:Here, we can also factor out
and then focus on the polynomial . -
Third Expression:This expression is already factored, showing
multiplied by two other factors. -
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!