In this page lcm and gcd solution1 we are going to see solution of some questions with detailed explanation.

(1) Find the LCM of each pair of the following polynomials

(i) x² - 5 x + 6 , x² + 4 x - 12 whose GCD is (x - 2)

**Solution:**

L.C.M **x** G.C.D = f (x) **x** g (x)

f (x) = x² - 5 x + 6

g (x) = x² + 4 x - 12

G.C.D = (x - 2)

L.C.M = [ f (x) **x** g (x) ]/G.C.D

= [ (x² - 5 x + 6) (x² + 4 x - 12) ]/(x - 2)

To simplify this we have to factorize these quadratic equations.So that we get

= [ (x - 2) (x - 3) (x + 6) (x - 2) ]/(x - 2)

= (x - 2) (x - 3) (x + 6)

(ii) x⁴ + 3 x³ + 6 x² + 5 x + 3 , x⁴ + 2 x² + x + 2 whose GCD is x² + x + 1

**Solution:**

L.C.M **x** G.C.D = f (x) **x** g (x)

f (x) = x⁴ + 3 x³ + 6 x² + 5 x + 3

g (x) = x⁴ + 2 x² + x + 2

G.C.D = x² + x + 1

L.C.M = [ f (x) **x** g (x) ]/G.C.D

= [ (x⁴ + 3 x³ + 6 x² + 5 x + 3) (x⁴ + 2 x² + x + 2) ]/(x² + x + 1)

To simplify this we have to use long division.

= (x² + 2 x + 3) (x⁴ + 2 x² + x + 2)

(iii) 2 x³ + 15 x² + 2 x - 35 , x⁴ + 8 x² + 4 x - 21 whose GCD is x + 7

**Solution:**

L.C.M **x** G.C.D = f (x) **x** g (x)

f (x) = 2 x³ + 15 x² + 2 x - 35

g (x) = x⁴ + 8 x² + 4 x - 21

G.C.D =x + 7

L.C.M = [ f (x) **x** g (x) ]/G.C.D

= [ (2 x³ + 15 x² + 2 x - 35) (x⁴ + 8 x² + 4 x - 21) ]/(x + 7)

To simplify this we have to use long division.

= (2 x² + x - 5) (x⁴ + 8 x² + 4 x - 21)

(iv) 2x³ - 3x² - 9x + 5 , 2x⁴ - x³ - 10 x² - 11 x + 8 whose GCD is 2x - 1

**Solution:**

L.C.M **x** G.C.D = f (x) **x** g (x)

f (x) = 2x³ - 3x² - 9x + 5

g (x) = 2x⁴ - x³ - 10 x² - 11 x + 8

G.C.D = 2x - 1

L.C.M = [ f (x) **x** g (x) ]/G.C.D

= [ (2x³ - 3x² - 9x + 5) (2x⁴ - x³ - 10 x² - 11 x + 8) ]/(2x - 1)

To simplify this we have to use long division.

= (x³ - 5x - 8) (2x³ - 3x² - 9x + 5)

lcm and gcd solution1 lcm and gcd solution1

- Back to worksheet
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