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The simplified rational expression of [tex] \frac{2x^2 - 7x - 4}{5x^2 - 16x + 3} \cdot \frac{2x^2 - 7x + 3}{5x^2 - 21x + 4} [/tex] is

A. [tex] \frac{4x}{5x - 1}, \, x \neq \frac{1}{5}, 3, 4 [/tex]
B. [tex] \frac{4x^2 - 1}{5x - 1}, \, x \neq \frac{1}{5}, 3, 4 [/tex]
C. [tex] \frac{4x^2 - 1}{25x^2 - 10x + 1}, \, x \neq \frac{1}{5}, 3, 4 [/tex]
D. [tex] \frac{4x^2 - 1}{25x^2 - 1}, \, x \neq \frac{1}{5}, 3, 4 [/tex]



Answer :

GinnyAnswer
To simplify the rational expression [tex]\(\frac{2 x^2-7 x-4}{5 x^2-16 x+3} \cdot \frac{2 x^2-7 x+3}{5 x^2-21 x+4}\)[/tex] and find the correct answer, we can break the process into several steps:

1. Write out the expressions for clarity:
[tex]\[ \text{Expression 1: } \frac{2 x^2 - 7 x - 4}{5 x^2 - 16 x + 3} \][/tex]
[tex]\[ \text{Expression 2: } \frac{2 x^2 - 7 x + 3}{5 x^2 - 21 x + 4} \][/tex]

2. Combine the expressions:
[tex]\[ \frac{2 x^2 - 7 x - 4}{5 x^2 - 16 x + 3} \cdot \frac{2 x^2 - 7 x + 3}{5 x^2 - 21 x + 4} \][/tex]

3. Multiply the numerators and the denominators:
[tex]\[ \frac{(2 x^2 - 7 x - 4)(2 x^2 - 7 x + 3)}{(5 x^2 - 16 x + 3)(5 x^2 - 21 x + 4)} \][/tex]

4. Simplify the combined expression:
The simplified form of the combined rational expression is:
[tex]\[ \frac{4 x^2 - 1}{25 x^2 - 10 x + 1} \][/tex]

5. Identify excluded values:
The values of [tex]\(x\)[/tex] that make any denominator in the original expressions equal to zero must be excluded. The original denominators are [tex]\(5 x^2 - 16 x + 3\)[/tex] and [tex]\(5 x^2 - 21 x + 4\)[/tex].

Solving [tex]\(5 x^2 - 16 x + 3 = 0\)[/tex] and [tex]\(5 x^2 - 21 x + 4 = 0\)[/tex] will yield the excluded values:
- For [tex]\(5 x^2 - 16 x + 3 = 0\)[/tex], solve to find roots: [tex]\(x = \frac{1}{5}, 3\)[/tex]
- For [tex]\(5 x^2 - 21 x + 4 = 0\)[/tex], solve to find roots: [tex]\(x = \text{root of the quadratic equation}\)[/tex]

The values that cause the quadratics to equal zero are determined as [tex]\(x = \frac{1}{5}, 3, 4\)[/tex].

Combining this information, we can conclude that the simplified expression is:
[tex]\[ \frac{4 x^2 - 1}{25 x^2 - 10 x + 1}, \quad x \neq \frac{1}{5}, 3, 4 \][/tex]

Therefore, the correct answer is:

C. [tex]\(\frac{4 x^2-1}{25 x^2-10 x+1}, x \neq \frac{1}{5}, 3, 4\)[/tex]

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