Question
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Given . Write in factored form (as a
product of linear factors). Be sure to write the full equation,
including .

Ask by Boyd Johnston. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

.

Solution

We are given
Step 1. Find a Rational Root
To factor as a product of linear factors, we start by finding a rational root. By the Rational Root Theorem, any rational root is a divisor of the constant term (45). Possible candidates include .
We test :
Thus, is a root and is a factor.
Step 2. Divide by
Divide by to find the quadratic factor.
Using synthetic division with root :
The quotient is
Thus, we have
Step 3. Factor the Quadratic Further
The quadratic can be factored over the complex numbers. Recall that
Thus,
Step 4. Write the Fully Factored Form
Therefore, the complete factorization of into linear factors (over the complex numbers) is:

Answered by UpStudy AI and reviewed by a Professional Tutor

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Extra Insights

To factor , we first look for rational roots using the Rational Root Theorem. Testing x = -3, we find that it is indeed a root. Thus, we perform polynomial long division of by and get .
Next, we find the roots of the quadratic using the quadratic formula, yielding complex roots . Therefore, the fully factored form of is:
So, the final factored form is:

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