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ALGEBRA E ALGEBRA Sol. p (x) q (x) - - = - = 25. x – 84 – 2xy + 4gx = (x2 - 9) (x - 3) Sol. x°- 8y - 2xy + 4y-x - (x + 3) (x-3) (x - 3) = x3 - (2y) - 2xy (x - 2y) = (x + 3) (x - 3)2 : (x-2y) [x² + 2xy + (2y)?] – 2xy (x - 2y) = x + 6x + 9 : (x - 2y) (x2 + 2xy + 4y4) - 2xy (x - 2y) = x2 + 3x + 3x + 9 = (x – 2y) (x + 2xy + 4y - 2xy) = x (x + 3) + 3 (x + 3) = (x + 3) (x + 3) = (x – 2y) (x + 4y?) - (x + 3)2 HCF of p(xt and g(x) = x + 3 G.C.D&L.C.M OF POLYNOMIALS Example : Find the LCM of p (x) = (x + 3) (x - 2)2 Greatest Common Divisor/Highest Common and q (x) = (x - 2) (x - 6). Factor (GCD/HCF) : The GCD of two polynomials p(x) Sol. p (2) = (x + 3) (x - 2)2 and g(x) is that common divisor which has the highet q (x (x-2)(x-6) degree among all common divisors and the coefficient HCF of p (x) and q (x) (x - 2) of the highest degree term be positive. Example : What is the HCF of (x + 4)2 (x-3) and p(x) xq (x) (x - 1) (x + 4) (x-3)2 ? LCM of p (x) and q (x) HCF Sol. p(x) = (x + 4)2 (x - 3)2 g(x) = (x - 1) (x + 4) (x-3)2 (x+3)(x - 2)2 x(x - 2)(x-6) We see that (x + 4) (x – 3)2 is such a polynomial (x-2) that is a common divisor and whose degree is high- est among all common divisors. = (x + 3) (x - 2)2 (x - 6) .:: HCF = (x + 4) (x - 3)2 Example : Find the LCM of 7x + 2x – 16x - 32 Lowest Common Multiple (LCM) : The LCM of and 3x2 - 2x-8. two polynomials p(x) and q(x) is a polynomial of Sol. p(x) = 7x3 + 2x? - 16x - 32 lowest degree of which p(x) and g(x) both are mul- .. p (2) = 7 (2)3 + 2 (2)2 – 16 (2) - 32 Example : Find the LCM of (x - 1) (x + 2)2 and = 56 + 8 - 32 - 32 = 0 (x - 1)3(x + 2). :: (x - 2), is a factor of q (x). Sol. p(x) = (x - 1) (x + 2)2 ton Y-23783 + 2x2 - 16x-32 17x2+ 16x + 16 Х - tiples. ...
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