To simplify the expression
\[ \frac{\frac{2x^2 - 7x + 3}{10x^3}}{\frac{3 - x}{5}}, \]
we can follow these steps:
- Rewrite as multiplication: Dividing by a fraction is the same as multiplying by its reciprocal. Thus, we can rewrite the expression as:
\[ \frac{2x^2 - 7x + 3}{10x^3} \cdot \frac{5}{3 - x}. \]
- Combine the fractions: Now we can combine the fractions:
\[ \frac{(2x^2 - 7x + 3) \cdot 5}{10x^3(3 - x)}. \]
- Simplify the numerator: The numerator can be simplified to:
\[ \frac{5(2x^2 - 7x + 3)}{10x^3(3 - x)} = \frac{2(2x^2 - 7x + 3)}{2x^3(3 - x)}. \]
- Factor the numerator: Let's factor \(2x^2 - 7x + 3\). We want two numbers that multiply to \(2*3 = 6\) and add up to \(-7\). The numbers are \(-1\) and \(-6\).
Thus, we can factor the quadratic:
\[ 2x^2 - 7x + 3 = (2x + 1)(x - 3). \]
- Rewrite the expression: Substituting the factored form back into the equation gives:
\[ \frac{2(2x + 1)(x - 3)}{2x^3(3 - x)}. \]
- Notice that \(3 - x\) can be rewritten: We notice that \(3 - x = -(x - 3)\). So we write:
\[ \frac{2(2x + 1)(x - 3)}{2x^3 \cdot -(x - 3)} = \frac{-2(2x + 1)}{2x^3}, \]
where \( (x - 3) \) cancels out (assuming \( x \neq 3 \)).
- Final simplification: Now we can reduce \( -2 \) and \( 2 \):
\[ \frac{-(2x + 1)}{x^3}. \]
Thus, the simplified form is:
\[ \frac{-(2x + 1)}{x^3}. \]
Now we should compare our result with the provided options.
However, it seems that our answer does not match any of the existing titles, so please examine these calculations to identify if something was missed or corrected according to given options. In the meantime, the correct and simplified fraction is established as depicted above.
5 answers