hcf of x^2-ax-(a+1) and ax^2-x-(a+1)
Answers
Given : x²-ax-(a+1) and ax²-x-(a+1)
To Find : HCF
Solution:
x²-ax-(a+1)
=x² - (a + 1)x + x - (a + 1)
= x(x - a - 1) + 1(x - a - 1)
= (x + 1)(x - (a + 1))
ax²-x-(a+1)
ax² + ax - (a + 1)x - (a + 1)
= ax(x + 1) - (a + 1)(x + 1)
= (x + 1) (ax - (a + 1))
Hence HCF = (x + 1)
HCF Highest common factor is highest power polynomial and
largest power numeric coefficient expression which can divide all the expressions.
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Answer:
x3 + (a +b)x2 + (ab + 1)x + b
= x3 + (a +b)x2 + abx + x + b
= x [x2 + (a + b)x + ab] + (x + b)
= x (x + a) (x + b) + (x + b)
= (x + b) [x (x + a) + 1]
= (x + b) (x2 + ax + a)
x 3 + 2ax 2 + (a 2 + 1)x + a
= x3 + 2ax2 + a 2 x + x + a
= x (x 2 + 2ax + a 2 ) + (x + a)
= x (x + a) (x + a) + (x + a)
= (x + a) [x (x + a) + 1]
= (x + a) (x 2 + ax + 1)
Common factor between the two polynomials = x 2 + ax + 1
∴ HCF = x 2 + ax +1