1. H.C.F. and L.C.M. of Polynomials

2. Do you know how to find the H.C.F. and L.C.M. of numbers?
Consider two numbers 60 and 72.
60 =
Express as products of prime factors.
72 =
H.C.F.=
= 12
For each COMMON prime factor, take the one with the smallest exponent.

3. Do you know how to find the H.C.F. and L.C.M. of numbers?
Consider two numbers 60 and 72.
60 =
Express as products of prime factors.
72 =
L.C.M.=
= 360
For each prime factor, take the one with the largest exponent.

4. How about finding the H.C.F. and L.C.M. of polynomials?
Similar techniques can be applied to find the H.C.F. and L.C.M. of polynomials.

5. For each COMMON factor, take the one with the lowest degree.
In fact,
The highest common factor (H.C.F.) of two or more polynomials is the polynomial of the highest degree which is a common factor of all the polynomials.
Consider two polynomials x2yz and x3y2.
x2yz = x2 y1 z1
x3y2 = x3 y2
H.C.F.=
x2 y1
= x2y
For each COMMON factor, take the one with the lowest degree.

6. For each factor, take the one with the highest degree.
The lowest common multiple (L.C.M.) of two or more polynomials is the polynomial of the lowest degree which is a common multiple of all the polynomials.
Again, consider two polynomials x2yz and x3y2.
x2yz = x2 y1 z1
x3y2 = x3 y2
L.C.M.=
x3 y2 z1
= x3y2z
For each factor, take the one with the highest degree.

7. Follow-up question
Find the H.C.F. and L.C.M. of 8x2y5z3, 6xy2 and 18x3yz2.
8x2y5z3 = x2 y z3
6xy2 = x y2
18x3yz2 = x y z2
∴ H.C.F. = 2 x y
= 2xy
L.C.M. = x3 y5 z3
= 72x3y5z3

8. How to find the H.C.F. and L.C.M. of x3 – 9x and 2x2 – 4x – 30?
We can factorize the polynomials first.

9. Find the H.C.F. and L.C.M. of x3 – 9x and 2x2 – 4x – 30.
x3 – 9x = x(x2 – 9)
= x(x2 – 32)
= x(x + 3)(x – 3)
 a2 – b2 ≡ (a + b)(a – b)
2x2 – 4x – 30 = 2(x2 – 2x – 15)
= 2(x + 3)(x – 5)
 By the cross method
∴ H.C.F. = x + 3
L.C.M. = 2 x (x + 3) (x – 3) (x – 5)
= 2x(x + 3)(x – 3)(x – 5)

10. Follow-up question
Find the H.C.F. and L.C.M. of 2x2 + 5x2 – 3 and 2x2 + 12x + 18.
2x2 + 5x – 3 = (x + 3)(2x – 1)
2x2 + 12x = 2(x2 + 6x + 9)
= 2(x + 3)2
∴ H.C.F. = 3 + x
L.C.M. =