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In this subtraction​ problem, the rational expression that follows the subtraction sign has a numerator with more than one term. Be careful with signs and find the difference.

StartFraction 2 q plus 5 Over 4 q squared plus 7 q minus 2 EndFraction minus StartFraction 5 q plus 1 Over 5 q squared plus 3 q minus 14 EndFraction

1 answer

To solve the subtraction problem involving rational expressions, we'll first need to simplify the two fractions and then perform the subtraction. The two fractions given are:

\[ \frac{2q + 5}{4q^2 + 7q - 2} - \frac{5q + 1}{5q^2 + 3q - 14} \]

Step 1: Factor the denominators if possible.

  1. Denominator \(4q^2 + 7q - 2\):

    • We can look for two numbers that multiply to \(4 \times -2 = -8\) and add to \(7\):
    • Those numbers are \(8\) and \(-1\).
    • We can rewrite the expression: \[ 4q^2 + 8q - q - 2 = (4q^2 + 8q) + (-q - 2) = 4q(q + 2) - 1(q + 2) = (4q - 1)(q + 2). \]

  2. Denominator \(5q^2 + 3q - 14\):

    • We look for two numbers that multiply to \(5 \times -14 = -70\) and add to \(3\):
    • Those numbers are \(10\) and \(-7\).
    • We can rewrite the expression: \[ 5q^2 + 10q - 7q - 14 = (5q^2 + 10q) + (-7q - 14) = 5q(q + 2) - 7(q + 2) = (5q - 7)(q + 2). \]

Step 2: Rewrite the subtraction with factored denominators.

Now we can rewrite the original subtraction with the factors of the denominators:

\[ \frac{2q + 5}{(4q - 1)(q + 2)} - \frac{5q + 1}{(5q - 7)(q + 2)}. \]

Step 3: Find a common denominator.

The common denominator will be \((4q - 1)(5q - 7)(q + 2)\).

Step 4: Rewrite each fraction with the common denominator.

  1. For the first fraction: \[ \frac{2q + 5}{(4q - 1)(q + 2)} \cdot \frac{(5q - 7)}{(5q - 7)} = \frac{(2q + 5)(5q - 7)}{(4q - 1)(5q - 7)(q + 2)}. \]
  2. For the second fraction: \[ \frac{5q + 1}{(5q - 7)(q + 2)} \cdot \frac{(4q - 1)}{(4q - 1)} = \frac{(5q + 1)(4q - 1)}{(4q - 1)(5q - 7)(q + 2)}. \]

Step 5: Combine the numerators and simplify.

Now we can do the subtraction:

\[ \frac{(2q + 5)(5q - 7) - (5q + 1)(4q - 1)}{(4q - 1)(5q - 7)(q + 2)}. \]

Step 6: Expand and simplify the numerator.

  1. Expand:

    • For the first term: \[ (2q + 5)(5q - 7) = 10q^2 - 14q + 25q - 35 = 10q^2 + 11q - 35. \]

    • For the second term: \[ (5q + 1)(4q - 1) = 20q^2 - 5q + 4q - 1 = 20q^2 - q - 1. \]

  2. Subtract: \[ (10q^2 + 11q - 35) - (20q^2 - q - 1) = 10q^2 + 11q - 35 - 20q^2 + q + 1 = -10q^2 + 12q - 34. \]

Final Expression

Thus, the difference we sought is: \[ \frac{-10q^2 + 12q - 34}{(4q - 1)(5q - 7)(q + 2)}. \]

We can also factor the numerator: \[ -2(5q^2 - 6q + 17). \] However, we will keep it in that form for simplicity.

Final answer:

\[ \frac{-10q^2 + 12q - 34}{(4q - 1)(5q - 7)(q + 2)}. \]