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Rational expressions Remember, a rational number is any number that can be a written in the form , where a and b are integers and b ≠ 0. b Numbers written in this form are often called fractions. In algebra, a rational expression is an algebraic fraction f(x) that can be written in the form , where f(x) and g(x) are polynomials and g(x) ≠ 0. g(x) For example, 3 3x+1 x3 2 x2 x2 2 x2 +3x 4 For which values of x are each of the above expressions undefined? 2 of 37 © Boardworks Ltd 2006 Rational expressions An algebraic fraction is undefined when the denominator is 0. So, 3 is undefined when x + 2 = That is, when x = –2. x+2 0. 3x+1 is undefined when x2 – 2 = 0. That is, when x = ±√2. x2 2 x3 2 is undefined when x2 + 3x – 4 = 0. x2 +3x 4 We can factorize this to give (x + 4)(x – 1) = 0. x3 2 So is undefined when x = –4 or x = 1. x2 +3x 4 3 of 37 © Boardworks Ltd 2006 Simplifying fractions by cancelling When the numerator and the denominator of a numerical fraction contain a common factor, the fraction can be simplified by cancelling. For example, consider the fraction 282 2 = 423 3 The highest common factor of 28 and 42 is _1_4_. This fraction can therefore be written in its simplest terms by dividing both the numerator and the denominator by 14. 4 of 37 © Boardworks Ltd 2006 Simplifying algebraic fractions by cancelling Algebraic fractions can be cancelled in the same way. For example, 2 3 6a = 6aa 3 3 = == 8a 8aaa 4a 4 When the numerator or the denominator contains more than one term, we have to factorize before cancelling. For example, 3pq Simplify 15p 9p2 3pq = 3pq = q 15p 9p2 3p(5 3p) 5 3p 5 of 37 © Boardworks Ltd 2006 Simplifying algebraic fractions by cancelling 3b 6 Simplify b2 2b 3b 6 = 3(b 2) = 3 b2 2b b(b 2) b x2 + x 2 Simplify 2 x 1 x2 + x 2 (x 1)(x 2) x2 2 = = x 1 (x 1)(x 1) x1 6 of 37 © Boardworks Ltd 2006
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