RATIONAL EXPRESSION-TIPS

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IMPORTANT TIPS

• ax + b (a ≠ 0) is a linear polynomial whereas ax² + bx + c (a ≠ 0)is a polynomial of second degree or a quadratic polynomial.



• If p(x) and q(x) are polynomials over R and q(x) ≠ 0, then p(x)/q(x) is called a rational expression.



• If the HCF of p(x) and q(x) is 1, i.e. there is nothing common between p(x) and q(x) except 1 , then p(x)/q(x) is called a rational expression in its REDUCED FORM.



• If p(x) = h(x) · p'(x) and q(x) = h(x) · q'(x)where h(x) is the HCF of p(x) and q(x), then the rational expression p(x)/q(x) can be reduced to p'(x)/q'(x), i.e. p'(x)/q'(x) is the reduced form of p(x)/q(x).



• To simplify p(x)/q(x), factorize both p(x) and q(x) into their prime factors; then cancel out like terms in both the numerator and the denominator.



• x + x = 2x (Some students write x + x = x². In fact, x · x = x²).



• 1 + 2 + x = 3 + x (Some students write 1 + 2 + x = 3x which is not true).



• Two polynomials can be added, subtracted, multiplied, and divided in the same manner as numbers.



• ADDITION:Two terms of same sign are added and the sign is retained. Thus, (-2) + (-3) = (-5) and (+4) + (+3) = (+7) or 4 + 3 = 7.
  
  The smaller term is subtracted from the bigger term if two terms have opposite signs and then the sign of the bigger term is retained in the result. Thus, (-7) + (+3) = (-4) and 7 + (-3) = 4.



• (a - b)² = (b - a)² BUT (a - b)³ ≠ (b - a)³. In general, (a - b)ⁿ = (b - a)ⁿ if n is even and (a - b)ⁿ = - (b - a)ⁿ if n is odd.



• x/x = 1. Similarly, (x + 2y)/(x + 2y) = 1. (Remember: x/x ≠ 0).



• (2x - 4)² ≠ 2(x - 2)² BUT (2x - 4)² = 4(x - 2)² and (2x - 4)³ = 8(x - 2)³.



• 16(x + 2)² ≠ (16x + 32)² BUT 16(x + 2)² = (4x + 8)² and 27(x - 3)³= (3x - 9)³.



• (2x + 10)/2 ≠ x + 10 (2 cancelled with 2 of 2x).
  
  (2x + 10)/2 ≠ 2x + 5 (2 cancelled with 2 of 10).
  
  Actually, (2x + 10)/2 = 2(x + 5)/2 = (x + 5).



• (4x + 5)/(2x + 5) ≠ 2.[You cannot cancel x + 5].



• While simplifying rational expression, strictly follow the rule of the order of operations: BODMAS → Brackets of Division, Multiplication, Addition and Subtraction.