To simplify the given rational expression and write it in lowest terms, we need to factor both the numerator and the denominator and then divide out any common factors.
The given expression is:
[tex]\[
\frac{x^2 - 12x + 27}{x^2 + 4x - 21}
\][/tex]
### Step 1: Factor the Numerator
First, we factor the numerator [tex]\(x^2 - 12x + 27\)[/tex]. We look for two numbers that multiply to [tex]\(27\)[/tex] and add up to [tex]\( -12\)[/tex].
- The factors of [tex]\(27\)[/tex] are: [tex]\(1 \cdot 27, 3 \cdot 9\)[/tex]
- The pair that adds up to [tex]\(-12\)[/tex] is [tex]\(-3\)[/tex] and [tex]\(-9\)[/tex].
So, we can write the numerator as:
[tex]\[
x^2 - 12x + 27 = (x - 3)(x - 9)
\][/tex]
### Step 2: Factor the Denominator
Next, we factor the denominator [tex]\(x^2 + 4x - 21\)[/tex]. We look for two numbers that multiply to [tex]\(-21\)[/tex] and add up to [tex]\(4\)[/tex].
- The factors of [tex]\(-21\)[/tex] are: [tex]\(-1 \cdot 21, -3 \cdot 7, 3 \cdot (-7)\)[/tex]
- The pair that adds up to [tex]\(4\)[/tex] is [tex]\(7\)[/tex] and [tex]\(-3\)[/tex].
So, we can write the denominator as:
[tex]\[
x^2 + 4x - 21 = (x + 7)(x - 3)
\][/tex]
### Step 3: Simplify the Rational Expression
Now we have the rational expression in factored form:
[tex]\[
\frac{(x - 3)(x - 9)}{(x + 7)(x - 3)}
\][/tex]
We notice that [tex]\(x - 3\)[/tex] is a common factor in both the numerator and the denominator, so we can cancel these out:
[tex]\[
\frac{(x - 3)(x - 9)}{(x + 7)(x - 3)} = \frac{x - 9}{x + 7}
\][/tex]
Thus, the simplified form of the given rational expression is:
[tex]\[
\frac{x - 9}{x + 7}
\][/tex]
So, the rational expression in its lowest terms is:
[tex]\[
\frac{x - 9}{x + 7}
\][/tex]
