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The Deep Dive
To simplify the expression
, we first rewrite the division as multiplication by the reciprocal:
Now, let’s factor each polynomial.
-
Factoring
: - Rearranging gives us
. - To factor this, we need two numbers that multiply to
(the product of and ) and add to . These numbers are and . - Thus, we can rewrite it as
.
- Rearranging gives us
-
Factoring
: - The product is
and we want the sum to be . The factor pair that works is and . - Rewriting gives
.
- The product is
-
Factoring
: - The product is
and we want the sum to be . The pair is and . - Rewriting gives
.
- The product is
-
Factoring
: - The product is
and we need . The correct pair is and . - Rearranging gives
.
- The product is
Putting all the factors together, the expression can now be simplified:
Now, we can cancel out like terms, specifically
in the numerator and denominator:
Now, we can express the answer clearly with factored polynomials or leave it in this form, depending on the problem requirement.
Thus, the final simplified result is:
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