L.C.M. of Polynomials by Factorization

Learn how to solve L.C.M. of polynomials by factorization splitting the middle term.

Solved examples on lowest common multiple of polynomials by factorization:

1. Find the L.C.M of m3 – 3m2 + 2m and m3 + m2 – 6m by factorization.

Solution:

First expression = m3 – 3m2 + 2m

                      = m(m2 – 3m + 2), by taking common β€˜m’

                      = m(m2 - 2m - m + 2), by splitting the middle term -3m = -2m - m

                      = m[m(m - 2) - 1(m - 2)]                     

                      = m(m - 2) (m - 1)                     

                      = m Γ— (m - 2) Γ— (m - 1)



Second expression = m3 + m2 – 6m

                          = m(m2 + m - 6) by taking common β€˜m’

                          = m(m2 + 3m – 2m - 6), by splitting the middle term m = 3m - 2m

                          = m[m(m + 3) - 2(m + 3)]

                          = m(m + 3)(m - 2)

                          = m Γ— (m + 3) Γ— (m - 2)

In both the expressions, the common factors are β€˜m’ and β€˜(m - 2)’; the extra common factors are (m - 1) in the first expression and (m + 3) in the 2nd expression.

Therefore, the required L.C.M. = m Γ— (m - 2) Γ— (m - 1) Γ— (m + 3)

                                         = m(m - 1) (m - 2) (m + 3)


2. Find the L.C.M of 3a3 - 18a2x + 27ax2, 4a4 + 24a3x + 36a2x2 and 6a4 - 54a2x2 by factorization.

Solution:

First expression = 3a3 -18a2x + 27ax2

                      = 3a(a2 - 6ax + 9x2), by taking common β€˜3a’

              = 3a(a2 - 3ax - 3ax + 9x2), by splitting the middle term - 6ax = - 3ax - 3ax

                      = 3a[a(a - 3x) - 3x(a - 3x)]                     

                      = 3a(a - 3x) (a - 3x)                     

                      = 3 Γ— a Γ— (a - 3x) Γ— (a - 3x)

Second expression = 4a4 + 24a3x + 36a2x2

                          = 4a2(a2 + 6ax + 9x2), by taking common β€˜4a2’

               = 4a2(a2 + 3ax + 3ax + 9x2), by splitting the middle term 6ax = 3ax + 3ax

                          = 4a2[a(a + 3x) + 3x(a + 3x)]

                          = 4a2(a + 3x) (a + 3x)

                          = 2 Γ— 2 Γ— a Γ— a Γ— (a + 3x) Γ— (a + 3x)

Third expression = 6a4 - 54a2x2

                      = 6a2(a2 - 9x2), by taking common β€˜6a2’

                      = 6a2[(a)2 - (3x)2), by using the formula of a2 – b2

                      = 6a2(a + 3x) (a - 3x), we know a2 – b2 = (a + b) (a – b)

                      = 2 Γ— 3 Γ— a Γ— a Γ— (a + 3x) Γ— (a - 3x)

The common factors of the above three expressions is β€˜a’ and other common factors of first and third expressions are β€˜3’ and β€˜(a - 3x)’.

The common factors of second and third expressions are β€˜2’, β€˜a’ and β€˜(a + 3x)’.

Other than these, the extra common factors in the first expression is β€˜(a - 3x)’ and in the second expression are β€˜2’ and β€˜(a + 3x)’

Therefore, the required L.C.M. = a Γ— 3 Γ— (a - 3x) Γ— 2 Γ— a Γ— (a + 3x) Γ— (a - 3x) Γ— 2 Γ— (a + 3x) = 12a2(a + 3x)2(a - 3x)2


More problems on L.C.M. of polynomials by factorization splitting the middle term:

3. Find the L.C.M. of 4(a2 - 4), 6(a2 - a - 2) and 12(a2 + 3a - 10) by factorization.

Solution:

First expression = 4(a2 - 4)

                      = 4(a2 - 22), by using the formula of a2 – b2

                      = 4(a + 2) (a - 2), we know a2 – b2 = (a + b) (a – b)

                      = 2 Γ— 2 Γ— (a + 2) Γ— (a - 2)

Second expression = 6(a2 - a - 2)

                          = 6(a2 – 2a + a - 2), by splitting the middle term – a = – 2a + a

                          = 6[a(a - 2) + 1(a - 2)]

                          = 6(a - 2) (a + 1)

                          = 2 Γ— 3 Γ— (a - 2) Γ— (a + 1)

Third expression = 12(a2 + 3a - 10)

                      = 12(a2 + 5a – 2a - 10), by splitting the middle term 3a = 5a – 2a

                      = 12[a(a + 5) - 2(a + 5)]

                      = 12(a + 5) (a - 2)

                      = 2 Γ— 2 Γ— 3 Γ— (a + 5) Γ— (a - 2)

In the above three expressions the common factors are 2 and (a - 2).

Only in the second expression and third expression the common factor is 3.

Other than these, the extra common factors are (a + 2) in the first expression, (a + 1) in the second expression and 2, (a + 5) in the third expression.

Therefore, the required L.C.M. = 2 Γ— (a - 2) Γ— 3 Γ— (a + 2) Γ— (a + 1) Γ— 2 Γ— (a + 5)

                                         = 12(a + 1) (a + 2) (a - 2) (a + 5)







8th Grade Math Practice

From L.C.M. of Polynomials by Factorization to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Area, Perimeter and Volume | Square, Rectangle, Cube,Cubo

    Jul 25, 25 12:21 PM

    In this worksheet on area perimeter and volume you will get different types of questions on find the perimeter of a rectangle, find the perimeter of a square, find the area of a rectangle, find the ar…

    Read More

  2. Worksheet on Volume of a Cube and Cuboid |The Volume of a RectangleBox

    Jul 25, 25 03:15 AM

    Volume of a Cube and Cuboid
    We will practice the questions given in the worksheet on volume of a cube and cuboid. We know the volume of an object is the amount of space occupied by the object.1. Fill in the blanks:

    Read More

  3. Volume of a Cuboid | Volume of Cuboid Formula | How to Find the Volume

    Jul 24, 25 03:46 PM

    Volume of Cuboid
    Cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid will have a length, breadth and height. Hence we can conclude that volume is 3 dimensional. To measur…

    Read More

  4. Volume of a Cube | How to Calculate the Volume of a Cube? | Examples

    Jul 23, 25 11:37 AM

    Volume of a Cube
    A cube is a solid box whose every surface is a square of same area. Take an empty box with open top in the shape of a cube whose each edge is 2 cm. Now fit cubes of edges 1 cm in it. From the figure i…

    Read More

  5. 5th Grade Volume | Units of Volume | Measurement of Volume|Cubic Units

    Jul 20, 25 10:22 AM

    Cubes in Cuboid
    Volume is the amount of space enclosed by an object or shape, how much 3-dimensional space (length, height, and width) it occupies. A flat shape like triangle, square and rectangle occupies surface on…

    Read More