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ALGEBRA! ALGEBRA (iii) If a polynomial p(x) is divided by (ax + b), then b remainder is the value of p(x) at x= --- a i.e. p ( ) a Note : It is to be noted that the degree of q(x) is less than that of plx) and polynomial of higher degree is always divided by a polynomial of lower degree. The operation of division ends when the remainder is ei- ther zero or the degree of remainder is less than that of divisor. In the above example, the quotient is x2 + 5x – 2 and remainder is zero. As the remainder is zero, (x-2) is a factor of x2 + 3x - 12x + 4. Example : Divide p (x) = x8 – 14x2 + 37x - 60 by g(x= x-2. b (.. ax + b=0X=--1 a (iv) If a polynomial p(x) is divided by b ax, the b remainder is the value of p(x) at x = 음 i.e. p ple a a x2_ 12x + 13 X3- 14x2 + 37x - 60 1. b-ax= 0 = x= bi a Example : Let p (x) = x4 3x + 2x + 5. Find re- mainder when p (x) is divided by (x - 1). P — Solution : X-2 x²– 2x² - + -12x2+ 37x - 60 -12x2+24x x3 + x2- 2x Solution : x-1 x4 + Oxl-3x2+ 2x + 5 \xt x² + + 13x - 60 13x - 26 + -34 x - 3x2 + 2x + 5 x - x² + - 2x2 + 2x + 5 2x2 + 2x + 5 Here, quotient = x2 - 12x + 13 and remainder =-34 Since remainder #0, then (x-2) is not a factor of x - 14x2 + 37x- 60. Remainder Theorem Lot Ahammominl of desroon1 and let abe Here, remainder = 5 Find the value of p(1) from the above example. ...
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