Question
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Answer
The polynomial
simplifies to
.
Solution
To factor the polynomial
, let’s follow these steps:
1. Combine Like Terms
First, simplify the expression by combining like terms:
So, the expression simplifies to:
2. Factor Out the Common Factor
Notice that each term in the simplified expression has a common factor of
. Factoring this out gives:
3. Attempt to Factor the Quadratic
Now, let’s attempt to factor the quadratic expression inside the parentheses:
To factor this, we look for two numbers that multiply to
(the constant term) and add up to
(the coefficient of the middle term). However, such real numbers do not exist because the discriminant of this quadratic equation is negative:
A negative discriminant indicates that the quadratic cannot be factored into real numbers.
4. Final Factored Form
Since the quadratic doesn’t factor nicely over the real numbers, the fully factored form of the original polynomial is:
Conclusion
The polynomial
simplifies and factors as:
This is the most factored form using real numbers.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Extra Insights
To factor the polynomial
, first combine like terms. The expression simplifies to
, or you can factor out a negative sign to get
. However, this quadratic does not factor neatly with rational numbers. Therefore, we can use the quadratic formula to find the roots instead:
where
and
. This helps find factors if necessary!
To rewrite it in factored form, you can express it as
, which shows the vertex form of the polynomial, but does not factor over the rationals. Remember, not all polynomials can be factored in simple forms!