Q. 1 F5.0( 2 Votes )

Write each polyno

Answer :

Given, p(x) = x2 + x – 1


Now, we need to write the given polynomial as a product of first degree polynomial and p(x) = 0


The given equation can be written as = x2 + x + (– 1) which is of the form x2 + (a + b)x + ab


a + b = 1 and ab = – 1


we know that (a + b)2 –(a – b)2 = 4ab


(a + b)2 – 4ab = (a – b)2


we know a + b = 1 and ab = – 1


1 – 4(– 1) = (a – b)2


(a – b)2 = 5


a – b =


Solving the equation both equation a + b and a – b we get as follows


a + b + a – b = 1 + √5


a = (1 + √5)


(a + b) – (a – b) = 1 + √5


b = (1 – √5)


x2 + x + (– 1) has factors (x + (1 + √5))(x + (1 – √5))


p(x) = 0 if (x + (1 + √5))is 0 and (x + (1 – √5))is 0


(x + (1 + √5)) = 0


x = – (1 + √5)


(– (1 + √5))2 + (– (1 + √5)) + (– 1) = 0


And


(x + (1 – √5)) = 0


x = – (1 – √5)


(– (1 – √5))2 + (– (1 – √5)) + (– 1) = 0


Hence, (x + (1 + √5))(x + (1 – √5)) are the first degree factors of the polynomial and – (1 + √5) and – (1 – √5) are the solution of the given polynomial x2 + x – 1


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