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.

Answers:

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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