Let f(x) and g(x) be two polymials.
We can find the LCM or GCD of the two polynomials using the relationship given below.
f (x) × g(x) = LCM × GCD
Problem 1 :
Find the LCM of the following polynomials whose GCD is (a - 2).
(a2 + 4a −12) and (a2 −5a + 6)
Solution :
Let f(x) = a2 + 4a −12, g(x) = a2 −5a + 6.
f(x) = a2 + 4a −12 = a2 + 6a - 2a −12 = a(a + 6) - 2(a + 6) f(x) = (a + 6)(a - 2) |
g(x) = a2 −5a + 6 = a2 - 2a - 3a + 6 = a(a - 2) - 3(a - 6) g(x) = (a - 3)(a - 2) |
GCD is (a -2)
f (x) × g(x) = LCM × GCD
LCM = [f(x) × g(x)] / GCD
LCM = [(a + 6)(a - 2) (a - 3)(a - 2)] / a -2
LCM = (a + 6)(a - 2) (a - 3)
Problem 2 :
Find the LCM of the following polynomials whose GCD is (x - 3a).
(x 4 -27a3x) and (x -3a)2
Solution :
Let f(x) = x 4 -27a3x, g(x) = (x -3a)2
f(x) = x(x3 - 27a3) f(x) = x(x3-(3a)3) f(x) = x(x-3a)(x2-x(3a)+(3a)2) f(x) = x (x- 3a)(x2-3ax+9a2) |
g(x) = (x -3a)2 |
GCD is (x -3a)
f (x) × g(x) = LCM × GCD
LCM = [f(x) × g(x)] / GCD
LCM = [x(x- 3a)(x2-3ax+9a2)(x -3a)2] / (x -3a)
LCM = x(x2-3ax+9a2)(x -3a)2
Problem 1 :
Find the GCD of the following polynomials.
12(x4 -x3) and 8(x4 −3x3 +2x2)
Given that LCM is 24x3(x -1)(x -2).
Solution :
Let f(x) = 12(x4 -x3), g(x) = 8(x4 −3x3 +2x2)
LCM = 24x3(x -1)(x-2)
f(x) = 12(x4 -x3) f(x) = 12x3(x - 1) |
g(x) = 8(x4 −3x3 +2x2) g(x) = 8x2(x2 - 3x + 2) |
GCD = 24x3(x -1)(x -2)
f (x) × g(x) = LCM × GCD
GCD = [f(x) × g(x)] / LCM
GCD = [12x3(x - 1) 8x2(x2 - 3x + 2)]/ 24x3(x -1)(x -2)
GCD = 4x2(x-1)
Problem 2 :
Find the GCD of the following polynomials.
(x3 + y3) and (x4 + x2y2 + y4)
Given that LCM is (x3 + y3)(x2 + xy + y2).
Solution :
Let f(x) = (x3 + y3), g(x) = (x4 + x2y2 + y4)
LCM is (x3 + y3)(x2 + xy + y2)
f(x) = (x3 + y3) |
g(x) = (x4 + x2y2 + y4) = (x2 + y2)2 - (xy)2 = (x2 + y2)2 - (xy)2 = (x2-xy+ y2 )(x2+ xy+ y2) |
LCM = (x3 + y3)(x2 + xy + y2)
f (x) × g(x) = LCM × GCD
GCD = [f(x) × g(x)] / LCM
GCD = [ (x3 + y3)(x2-xy+ y2 )(x2+ xy+ y2)] / (x3 + y3)(x2 + xy + y2)
GCD = (x2 - xy + y2)
Problem 3 :
LCM and GCD of the two polynomials p(x) and q(x) and the polynomial p(x) are given below. Find q(x).
LCM = a3 −10a2 +11a + 70
GCD = a - 7
p(x) = a2 −12a + 35
Solution :
p (x) × q(x) = LCM × GCD
Then,
q(x) = (LCM × GCD) / p (x)
= (a3 −10a2 +11a + 70)( a - 7)/(a2 −12a + 35)
q(x) = (a + 2) (a - 7)
Problem 4 :
LCM and GCD of the two polynomials p(x) and q(x) and the polynomial q(x) are given below. Find p(x).
LCM = (x2 +y2)(x4 +x2y2+y4)
GCD = (x2 -y2)(x4 −y4)
q(x) = (x2 +y2 −xy)
Solution :
p (x) × q(x) = LCM × GCD
Then,
p(x) = (LCM × GCD)/q(x)
= (x2 + y2)(x4 + x2y2 + y4)(x2 - y2) / (x4 − y4)(x2 + y2− xy)
= (x4 - y4)(x4 + x2y2 + y4) / (x4 − y4)(x2 + y2− xy)
= (x4 + x2y2 + y4) / (x2 + y2− xy)
= [(x2 + y2)2 - (xy)2] / (x2 + y2− xy)
= (x2 - xy + y2)(x2 + xy + y2) / (x2 − xy + y2)
= (x2+ xy+ y2)
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