Worksheet on L.C.M. of Polynomials

Practice the worksheet on L.C.M. of polynomials. The questions are based on finding the lowest common multiple of two or more polynomials.

We know, to find the lowest common multiple (L.C.M.) of two or more than two polynomials is the polynomial of lowest measures (or dimensions) which is divisible by each of the polynomials without remainder.

For example:The lowest common multiple of 3m2 + 9mn, 2m3 – 18mn2 and m3 + 6m2n + 9mn2

It will be easy to pick out the common multiples if the polynomials are arranged as follows;

3m2 + 9mn = 3m(m + 3n); [taking the common factor 3m from both the terms].

2m3 – 18mn2 = 2m(m + 3n) (m - 3n); [taking the common factor 2m from both the terms and then factor using the formula of a2 – b2].

m3 + 6m2n + 9mn2 = m(m + 3n)2; [taking the common factor m from each term and then factor using the formula of (a + b)2]

Therefore the L.C.M. of 3m2 + 9mn, 2m3 – 18mn2 and m3 + 6m2n + 9mn2 = 6m(m + 3n)2 (m - 3n).

1. Find the lowest common multiple (L.C.M.) of the two polynomials:

(i) m and m2 + m

(ii) n2 and n2 – 3n

(iii) 3k2 and 4k2 + 8k

(iv) 21p3 and 7p2(p + 1)

(v) p2 – 1 and p2 + p

(vi) x2 + xy and y2 + xy

(vii) 4p2q – q and 2p2 + p

(viii) 6k2 – 2k and 9k2 – 3k

(ix) a2 + 2b and a2 + 3a + 2

(x) d2 – 3d + 2 and d2 - 1

2. Find the lowest common multiple (L.C.M.) of the three polynomials:

(i) a2 – a – 6, a2 + a – 2 and a2 – 4a + 3

(ii) 3n2 – n – 14, 3n2 – 13n + 14 and n2 - 4

(iii) (2x^2 – 3xy)2, (4x – 6y)3 and 8x3 – 27y3

(iv) a2 + a – 20, a2 – 10a + 24 and a2 – a - 30

(v) a4 + a2b2 + b4, a3b + b4 and (a3 – ab)3

Answers for the worksheet on L.C.M. of polynomials are given below to check the exact answers of the above questions.

1. (i) m(1 + m)

(ii) n2(n – 3)

(iii) 12k2(k + 2)

(iv) 21p3(p + 1)

(v) p(p + 1) (p – 1)

(vi) xy(x + y)

(vii) pq(2p + 1) (2p – 1)

(viii) 6k(3k – 1)

(ix) a(a + 1) (a + 2)

(x) (d + 1) (d – 1) (d – 2)

2. (i) (a - 3) (a – 1) (a + 2)

(ii) (n + 2) (n – 2) (3n – 7)

(iii) 8x2(2x – 3y)2 (8x3 – 27y3)

(iv) (a + 5) (a – 4) (a – 6)

(v) a3b(a6 – b6) (a – b)2

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