Question

1. Simplify the following as a quotient of two polynomials in the simplest form.

Answer 1

SOLUTION IS IN THE ATTACHMENT

RATIONAL EXPRESSIONS :
A rational number is defined as a quotient a/b, of two integers a and b and b ≠0.A rational expression is a quotient p(x) /q(x) of two Polynomials p(x) and q(x) ,where q(x) ≠0.
•When the numerator and denominator of a rational expression do not have any common factor except 1, the rational expression is said to be expressed in the lowest terms.
•To reduce a rational expression to the lowest terms, factorise the numerator and the denominator and cancel the factors which are common to both.

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Answer 2

Solution:-

given by:-
$\frac{ {x}^{2} - x - 6}{ {x}^{2} - 9 } + \frac{ {x}^{2} + 2x - 24 }{ {x}^{2} - x - 12} \\ \frac{ {x}^{2} - 3x + 2x - 6 }{(x - 3)(x + 3)} + \frac{ {x}^{2} + 6x - 4x - 24 }{ {x}^{2} - 4x + 3x - 12 } \\ \frac{x(x - 3) + 2(x - 3)}{(x - 3)(x + 3)} + \frac{x(x + 6) - 4(x + 6)}{x(x - 4) + 3(x - 4)} \\ \frac{(x - 3)(x + 2)}{(x - 3)(x + 3)} + \frac{(x + 6)(x - 4)}{(x - 4)(x + 3)} \\ \frac{x + 2}{x + 3} + \frac{x + 6}{x + 3} \\ \frac{x + 2 + x + 6}{x + 3} \\ \frac{2x + 8}{x + 3} \\ \frac{2(x + 4)}{(x + 3)} ans$
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