Final answer:
To multiply and simplify the rational expression, factor each quadratic, multiply them across, and cancel the common factors to simplify the expression. The final simplified form is 1/((x+8)(x+7)).
Step-by-step explanation:
This is done by factoring the numerator and denominator of each fraction and then multiplying the numerators together and the denominators together. It is similar to multiplication of whole numbers in that you multiply across the numerators and denominators and simplify by canceling any common factors.
To begin this process, we first factorize each quadratic in the expression:
- The first quadratic (x²-25) factors to (x+5)(x-5).
- The second quadratic (x²-64) factors to (x+8)(x-8).
- The third quadratic (x²-3x-40) factors to (x-8)(x+5).
- The fourth quadratic (x²+2x-35) factors to (x+7)(x-5).
Now, we multiply the factored forms of the numerators together and the factored forms of the denominators together and then cancel any common terms:
((x+5)(x-5))/((x+8)(x-8)) * ((x-8)(x+5))/((x+7)(x-5))
Simplifying, we observe that x+5 and x-5 cancel out, as do x-8. So, we are left with:
1/((x+8)(x+7))