Sure, let's simplify the given expression step-by-step.
We are given the expression:
[tex]\[ \frac{x-1}{3x^2 - 11x + 10} \][/tex]
Step 1: Factor the denominator [tex]\(3x^2 - 11x + 10\)[/tex]:
To factorize [tex]\(3x^2 - 11x + 10\)[/tex], we look for two numbers that multiply to [tex]\(3 \cdot 10 = 30\)[/tex] and add to [tex]\(-11\)[/tex]. Those numbers are [tex]\(-5\)[/tex] and [tex]\(-6\)[/tex]. So, we can rewrite the quadratic expression using these two numbers:
[tex]\[ 3x^2 - 11x + 10 = (3x^2 - 6x) - (5x - 10) \][/tex]
Next, we factor by grouping:
[tex]\[ 3x(x - 2) - 5(x - 2) = (3x - 5)(x - 2) \][/tex]
So the denominator can be rewritten as:
[tex]\[ 3x^2 - 11x + 10 = (3x - 5)(x - 2) \][/tex]
Step 2: Rewrite the whole fraction with the factored denominator:
[tex]\[ \frac{x - 1}{(3x - 5)(x - 2)} \][/tex]
In this case, the numerator [tex]\(x - 1\)[/tex] does not have any common factors with the denominator [tex]\((3x - 5)(x - 2)\)[/tex]. Therefore, the simplified form of the expression is:
[tex]\[ \frac{x - 1}{(3x - 5)(x - 2)} \][/tex]
