Final answer:
The question involves simplifying a complex algebraic fraction by factorizing polynomials and cancelling common terms. The process includes factoring quadratic expressions and looking for differences of squares, followed by subtraction of the simplified terms.
Step-by-step explanation:
To fully simplify the given algebraic expression, we'll look for factors that the terms have in common and cancel them out wherever possible. The expression in question is a complex fraction involving polynomial division and a subtraction operation:
(x2+12x+27)/(42x-6x2)*(49-x2)/(x2+16x+63) - (x2+8x+15)/(100-4x2)
To simplify each fraction, we should factorise the polynomials where possible and reduce them by cancelling common terms. The expression can be simplified further by seeking factorization such as (x + a)(x + b) for quadratic expressions and checking for differences of squares such as a2 - b2 = (a + b)(a - b). We should also remember to check for common numerical factors that can be simplified, particularly in expressions with complex coefficients.
As a step-by-step guide, we would:
- Factorise each of the numerators and denominators where possible.
- Cancel out common terms between numerators and denominators.
- Once the fractions have been simplified by cancellation of terms, carry out the subtraction.
- Finally, check to make sure the resulting expression is fully simplified.
Keep in mind that to truly simplify the expression, steps such as distribution, combining like terms, and checking for common factors are essential and must be followed thoroughly.