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Simplify the expression:

[tex]\[
\frac{x-1}{3x^2 - 11x + 10}
\][/tex]


Sagot :

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]
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