Studdy Solution
STEP 1
1. We need to simplify the given expression.
2. The expression involves adding two rational expressions.
3. We should find a common denominator to add the fractions.
4. Factoring polynomials will help in simplifying the expression.
STEP 2
1. Factor the denominators and numerators where possible.
2. Determine the least common denominator (LCD).
3. Rewrite each fraction with the LCD.
4. Add the fractions.
5. Simplify the resulting expression.
STEP 3
Factor the denominators and numerators:
- The first denominator: x2−25 is a difference of squares:
x2−25=(x−5)(x+5)
- The second denominator: 2x2+9x−5 can be factored by finding two numbers that multiply to 2×−5=−10 and add to 9. These numbers are 10 and −1:
2x2+9x−5=(2x−1)(x+5)
- The first numerator: 3x+15 can be factored by taking out the common factor 3:
3x+15=3(x+5)
- The second numerator: 4x2−1 is a difference of squares:
4x2−1=(2x−1)(2x+1)
STEP 4
Determine the least common denominator (LCD) from the factored denominators:
- The LCD is (x−5)(x+5)(2x−1).
STEP 5
Rewrite each fraction with the LCD:
- First fraction: (x−5)(x+5)3(x+5) becomes (x−5)(x+5)(2x−1)3(x+5)(2x−1).
- Second fraction: (2x−1)(x+5)(2x−1)(2x+1) becomes (x−5)(x+5)(2x−1)(2x−1)(2x+1)(x−5).
STEP 6
Add the fractions:
Combine the numerators over the common denominator:
(x−5)(x+5)(2x−1)3(x+5)(2x−1)+(2x−1)(2x+1)(x−5)
STEP 7
Expand and simplify the numerators:
- Expand 3(x+5)(2x−1):
3(x+5)(2x−1)=3(2x2−x+10x−5)=6x2+27x−15
- Expand (2x−1)(2x+1)(x−5):
(2x−1)(2x+1)=4x2−1
(4x2−1)(x−5)=4x3−20x2−x+5
Combine the expanded numerators:
6x2+27x−15+4x3−20x2−x+5=4x3−14x2+26x−10
STEP 8
Simplify the resulting expression:
The simplified expression is:
(x−5)(x+5)(2x−1)4x3−14x2+26x−10
