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