From beginner to expert, IDNLearn.com has answers for everyone. Get accurate and timely answers to your queries from our extensive network of experienced professionals.
Sagot :
To factor the polynomial [tex]\( x^3 - 2x^2 + x - 2 \)[/tex] completely, we need to follow several steps. Let's start by finding its roots.
### Step 1: Rational Root Theorem
The Rational Root Theorem states that any rational root of a polynomial equation, say [tex]\( p(x) = 0 \)[/tex], is a fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
For the polynomial [tex]\( x^3 - 2x^2 + x - 2 \)[/tex]:
- The constant term is [tex]\(-2\)[/tex].
- The leading coefficient is [tex]\(1\)[/tex].
The possible rational roots are [tex]\( \pm 1, \pm 2 \)[/tex].
### Step 2: Test the Potential Rational Roots
We will now test these potential roots by substituting them into the polynomial and checking if it equals zero.
#### Testing [tex]\( x = 1 \)[/tex]:
[tex]\[ 1^3 - 2(1)^2 + 1 - 2 = 1 - 2 + 1 - 2 = -2 \][/tex]
[tex]\[ \Rightarrow x = 1 \text{ is not a root.} \][/tex]
#### Testing [tex]\( x = -1 \)[/tex]:
[tex]\[ (-1)^3 - 2(-1)^2 + (-1) - 2 = -1 - 2 - 1 - 2 = -6 \][/tex]
[tex]\[ \Rightarrow x = -1 \text{ is not a root.} \][/tex]
#### Testing [tex]\( x = 2 \)[/tex]:
[tex]\[ 2^3 - 2(2)^2 + 2 - 2 = 8 - 8 + 2 - 2 = 0 \][/tex]
[tex]\[ \Rightarrow x = 2 \text{ is a root.} \][/tex]
### Step 3: Polynomial Division
Since [tex]\( x = 2 \)[/tex] is a root, we know that [tex]\( x - 2 \)[/tex] is a factor of the polynomial. We will now divide [tex]\( x^3 - 2x^2 + x - 2 \)[/tex] by [tex]\( x - 2 \)[/tex] using synthetic division or long division.
#### Synthetic Division:
[tex]\[ \begin{array}{r|rrrr} 2 & 1 & -2 & 1 & -2 \\ \hline & & 2 & 0 & 2 \\ \hline & 1 & 0 & 1 & 0 \\ \end{array} \][/tex]
The result of the division is [tex]\( x^2 + 1 \)[/tex], so:
[tex]\[ x^3 - 2x^2 + x - 2 = (x - 2)(x^2 + 1) \][/tex]
### Step 4: Factor [tex]\( x^2 + 1 \)[/tex]
Notice that [tex]\( x^2 + 1 \)[/tex] can be factored further over the complex numbers:
[tex]\[ x^2 + 1 = (x + i)(x - i) \][/tex]
### Conclusion
Putting it all together, the complete factorization of [tex]\( x^3 - 2x^2 + x - 2 \)[/tex] over the complex numbers is:
[tex]\[ x^3 - 2x^2 + x - 2 = (x - 2)(x + i)(x - i) \][/tex]
Thus, the correct answer is [tex]\( \boxed{A.(x-2)(x+i)(x-i)} \)[/tex].
### Step 1: Rational Root Theorem
The Rational Root Theorem states that any rational root of a polynomial equation, say [tex]\( p(x) = 0 \)[/tex], is a fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
For the polynomial [tex]\( x^3 - 2x^2 + x - 2 \)[/tex]:
- The constant term is [tex]\(-2\)[/tex].
- The leading coefficient is [tex]\(1\)[/tex].
The possible rational roots are [tex]\( \pm 1, \pm 2 \)[/tex].
### Step 2: Test the Potential Rational Roots
We will now test these potential roots by substituting them into the polynomial and checking if it equals zero.
#### Testing [tex]\( x = 1 \)[/tex]:
[tex]\[ 1^3 - 2(1)^2 + 1 - 2 = 1 - 2 + 1 - 2 = -2 \][/tex]
[tex]\[ \Rightarrow x = 1 \text{ is not a root.} \][/tex]
#### Testing [tex]\( x = -1 \)[/tex]:
[tex]\[ (-1)^3 - 2(-1)^2 + (-1) - 2 = -1 - 2 - 1 - 2 = -6 \][/tex]
[tex]\[ \Rightarrow x = -1 \text{ is not a root.} \][/tex]
#### Testing [tex]\( x = 2 \)[/tex]:
[tex]\[ 2^3 - 2(2)^2 + 2 - 2 = 8 - 8 + 2 - 2 = 0 \][/tex]
[tex]\[ \Rightarrow x = 2 \text{ is a root.} \][/tex]
### Step 3: Polynomial Division
Since [tex]\( x = 2 \)[/tex] is a root, we know that [tex]\( x - 2 \)[/tex] is a factor of the polynomial. We will now divide [tex]\( x^3 - 2x^2 + x - 2 \)[/tex] by [tex]\( x - 2 \)[/tex] using synthetic division or long division.
#### Synthetic Division:
[tex]\[ \begin{array}{r|rrrr} 2 & 1 & -2 & 1 & -2 \\ \hline & & 2 & 0 & 2 \\ \hline & 1 & 0 & 1 & 0 \\ \end{array} \][/tex]
The result of the division is [tex]\( x^2 + 1 \)[/tex], so:
[tex]\[ x^3 - 2x^2 + x - 2 = (x - 2)(x^2 + 1) \][/tex]
### Step 4: Factor [tex]\( x^2 + 1 \)[/tex]
Notice that [tex]\( x^2 + 1 \)[/tex] can be factored further over the complex numbers:
[tex]\[ x^2 + 1 = (x + i)(x - i) \][/tex]
### Conclusion
Putting it all together, the complete factorization of [tex]\( x^3 - 2x^2 + x - 2 \)[/tex] over the complex numbers is:
[tex]\[ x^3 - 2x^2 + x - 2 = (x - 2)(x + i)(x - i) \][/tex]
Thus, the correct answer is [tex]\( \boxed{A.(x-2)(x+i)(x-i)} \)[/tex].
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.
