Category: Mathematics

To calculate the Lowest Common Multiple of polynomials, you must find the factors of each polynomial and multiply the least occurred term. Check the steps to solve the L.C.M of polynomials and example questions in the further sections of this page.

Steps to Calculate L.C.M of Polynomials

Follow the detailed procedure to get the lowest common factor of polynomials mentioned below. These guidelines make it easy for you during the calculations.

• Get the factors of each polynomial.
• Identify common factors.
• Multiply the common factors and extra factors.

Some Examples on the Lowest Common Multiple of Polynomials

Example 1:

Find the lowest common multiple of x² − x, x², (x − 1)².

Solution:

Factorizing x² − x by taking the common factor ‘x’ we get,

= x(x − 1)

Also, factorizing x² we get

= x * x

Also, factorizing (x − 1)² we get,

= (x – 1) * (x – 1)

Therefore, the L.C.M. of x² − x, x², (x − 1)² is x²(x – 1)².

Example 2:

Find the L.C.M of n² − 3n + 2, n² − 4.

Solution:

Factorizing n² − 3n + 2 by splitting the middle term.

= n² – 2n – n + 2

= n (n – 2) -1 (n – 2)

= (n – 1) (n – 2)

Factorizing n² − 4 by using a² – b² formula.

= n² – 2²

= (n – 2) (n + 2)

Therefore, the L.C.M. of n² − 3n + 2, n² − 4 is (n – 2) (n + 2) (n – 1).

Example 3:

Find the least common multiple of 8x − 4, 6x² + x − 2.

Solution:

Factorizing 8x – 4 by taking common ‘4’.

= 4(2x – 1)

Factorizing 6x² + x – 2 by splitting the middle term.

= 6x² + 4x – 3x – 2

= 2x(3x + 2) – 1(3x + 2)

= (3x + 2) (2x – 1)

Therefore, the L.C.M of 8x − 4, 6x² + x − 2 is 4(2x-1) (3x+2).

You can compute the highest common factor or the greatest common factor of any number of polynomials by reading this article. Here we are giving the step by step explanation to get the H.C.F of polynomials along with the solved examples. Go through the below sections to solve the questions easily.

Step By Step Process to get G.C.F of Polynomials

Students can check out the below sections to find the detailed procedure of calculating the greatest common factor of polynomials,

• First of all, find the factors of polynomials.
• Identify the expression which is occurring more times.
• Separate the common factors from given polynomials and multiply them.

Example Questions on H.C.F of Polynomials

Example 1:

Find the Highest Common Factor of the polynomials x² – 6x + 9 and x² – 9.

Solution:

Factorizing x² – 6x + 9 by using the identities (a – b)², we get

(x)² – 2(x)(3) + (3)²

= (x – 3)²

= (x – 3) (x – 3)

Also, factorizing x² – 9, we get

(x)² – (3)², by using the identities of a² – b².

= (x + 3) (x – 3)

Therefore, H.C.F. of x² – 6x + 9 and x² – 9 is (x – 3).

Example 2:

Find the H.C.F of (a + b)² and (a² – b²).

Solution:

Factors of (a + b)² = (a + b) (a + b)

Factors of (a² – b²) = (a + b) (a – b)

The common factor is (a + b)

Therefore, the highest common factor of (a + b)² and (a² – b²) is (a + b).

Example 3:

Find the highest common factor of polynomials x² + 15x + 56, x² + 5x – 24 and x² + 8x.

Solution:

Factorizing x² + 15x + 56 by splitting the middle term, we get

= x² + 8x + 7x + 56

= x(x + 8) + 7(x + 8)

= (x + 7) (x + 8)

Also, factorizing x² + 5x – 24, we get

= x² + 8x – 3x – 24

= x (x + 8) – 3 (x + 8)

= (x + 3) (x + 8)

Factoring x² + 8x by taking x common.

= x (x + 8)

In all three polynomials the common factor is (x + 8)

Therefore, H.C.F. of x² + 15x + 56, x² + 5x – 24 and x² + 8x is (x + 8).

You can calculate the least common multiple of two or more monomials by using the factorization method. Get the factors of numerical coefficients and literal coefficients. From the factors, observe the least common multiples and multiply them. And find the product of the lowest common multiples of numerical and literal coefficients of monomials. Have a look at the sample example questions on Lowest Common Multiple of Monomials by Factorization in the below sections.

Example Questions on LCM of Monomials

Example 1:

What is the least common multiple of 5x³y² and 7x²y³?

Solution:

5x³y² = 5 * x * x * x * y * y

7x²y³ = 7 * x * x * y * y * y

From the resolved factors of the above two monomials, the common factors are x, x, y, y.

The extra factors in the first monomial are 5 and in the second monomial are 7, y.

Therefore, the required L.C.M. = Common factors among two monomials × Extra common factors among two monomials.

= (x * x * y * y) x (5 * 7 * y)

= 35x²y³

Hence, the lowest common multiple of the monomials 5x³y² and 7x²y³ = 35x²y³.

Example 2:

Find the L.C.M of 14xy, 21y², 28y.

Solution:

The L.C.M. of numerical coefficients = The L.C.M. of 14, 21, and 28.

Since, 14 = 2 * 7, 21 = 7 * 3, and 28 = 2 * 2 * 7

Therefore, the L.C.M. of 14, 21 and 28 is 2 * 7 * 2 * 3 = 84

The L.C.M. of literal coefficients = The L.C.M. of xy, y², and y = xy²

Since, in xy, y², and y

The highest power of x is 1.

The highest power of y is 2.

Therefore, the L.C.M. of xy, y², and y = xy².

Thus, the L.C.M. of 14xy, 21y², 28y

= The L.C.M. of numerical coefficients × The L.C.M. of literal coefficients

= 84 × (xy²)

= 84xy²

Example 3:

Find the L.C.M of 27u⁴, 18u², 27u².

Solution:

The L.C.M. of numerical coefficients = The L.C.M. of 27, 18, and 27.

Since, 27 = 3 * 3 * 3, 18 = 3 * 3 * 2, 27 = 3 * 3 * 3

Therefore, the L.C.M. of 27, 18, and 27 is 2 * 3 * 3 * 3 = 54

The L.C.M. of literal coefficients = The L.C.M. of u⁴, u², u² = u⁴

Since, in u⁴, u², u²

The highest power of u is 4.

Therefore, the L.C.M. of u⁴, u², u² = u⁴.

Thus, the L.C.M. of 27u⁴, 18u², 27u²

= The L.C.M. of numerical coefficients × The L.C.M. of literal coefficients

= 54 × (u⁴)

= 54u⁴

The lowest common multiple of monomials or L.C.M of monomials is the least occurred term in the monomials. Find the numerical coefficients LCM, literal coefficients LCM, and multiply them to get the result. You can check out the following sections to get the simple steps to calculate the least common multiple of two or monomials. Also, find the solved example questions for a better understanding of the concept.

How to Find L.C.M of Monomials?

Learn about how to get the lowest common multiple of monomials by using the steps mentioned below. Follow these easy to use steps to get the answer easily.

• Find the factors of given monomials numerical coefficients.
• And get the least common multiple from those factors.
• Get the highest power of each variable from the monomials.
• Calculate the LCM of literal coefficients.
• Multiply the LCM of numerical coefficients and literal coefficients.

Examples on Lowest Common Multiple of Monomials

Example 1:

Find the LCM of 12x²y³z and 18xy²z.

Solution:

The L.C.M of Numerical coefficients = The L.C.M. of 12, 18.

Since, 12 = 2 * 2 * 3 and 18 = 2 * 3 * 3

Therefore, the LCM of 12, 18 is 2 * 2 * 3 * 3= 36

The L.C.M. of literal coefficients = The L.C.M. of x²y³z, xy²z = x²y³z

Since in x²y³z and xy²z

The highest power of x is 2.

The highest power of y is 3.

The highest power of z is 1.

Therefore, the L.C.M. of x²y³z, xy²z = x²y³z

Thus, the L.C.M. of 12x²y³z and 18xy²z = The L.C.M. of numerical coefficients × The L.C.M. of literal coefficients

= 36 * x²y³z

= 36x²y³z

Example 2:

Find the L.C.M of 26p⁴q²r³ and 16p³q²r².

Solution:

The L.C.M. of numerical coefficients = The L.C.M. of 26, 16

Since, 26 = 2 * 13, 16 = 2 * 2 * 2 * 2

Therefore, the L.C.M. of 26, 16 is 2 * 13 * 8 = 206.

The L.C.M. of literal coefficients = The L.C.M. of p⁴q²r³, p³q²r²

Since in p⁴q²r³ and p³q²r²,

The highest power of p is 4.

The highest power of q is 2.

The highest power of r is 3.

Therefore, the L.C.M. of p⁴q²r³ and p³q²r² = p⁴q²r³

Thus, the L.C.M of 26p⁴q²r³ and 16p³q²r² = The L.C.M. of numerical coefficients × The L.C.M. of literal coefficients

= 206 * p⁴q²r³

= 206p⁴q²r³

Example 3:

Find the LCM of 24y³x, 40x³y.

Solution:

The L.C.M. of numerical coefficients = The L.C.M. of 24 and 40.

Since, 24 = 2 * 2 * 2 * 3 and 40 = 2 * 2 * 2 * 5

Therefore, the L.C.M. of 24 and 40 is = 2 * 2 * 2 * 3 * 5 = 120

The L.C.M. of literal coefficients = The L.C.M. of y³x and x³y = x³y³

Since, in y³x and x³y,

The highest power of x is 3.

The highest power of y is 3.

Therefore, the L.C.M. of y³x and x³y = x³y³.

Thus, the L.C.M. of 24y³x, 40x³y

= The L.C.M. of numerical coefficients × The L.C.M. of literal coefficients

= 120 × (x³y³)

= 120x³y³.

We are using the factorization method to calculate the highest common factor of monomials. Factorization is the process of breaking an entity into a product of another entity which when multiplied together gives the original entity. Get the example questions on finding the greatest common factor or Highest Common Factor of Monomials by Factorization.

HCF of Monomials Solved Examples

Example 1.

Find the H.C.F. of the monomials 16x²y³and 22x²y.

Solution:

The H.C.F. of numerical coefficients = The H.C.F. of 16, 22

Since, 16 = 2 * 2 * 2 * 2, 22 = 2 * 11

Therefore, the H.C.F. of 16, 22 is 2.

Now, the variables x and y are present in all the quantities. Out of these the highest common power of x is 2 and the highest common power if y is 1.

Therefore, the required H.C.F. = 2x²y¹ = 2x²y

The method by which the Highest common factor of the monomials is determined can be formulated as follows:

(i) The G.C.F. of the numerical coefficients are to be determined at first.

(ii) Then the variables are to be written beside the coefficient with their greatest common power or highest common power.

Example 2.

Find the Greatest Common Factor of the monomials 90x⁴y²z³, 54x²y³z⁴, and 18x³y²z³.

Solution:

The H.C.F. of numerical coefficients = The H.C.F. of 90, 54, and 18

Since, 90 = 2 * 5 * 9, 54 = 2 * 3 * 9, 18 = 2 * 9

Therefore, the H.C.F. of 90, 54, and 18 is 2 * 9 = 18.

Now, the variables x, y, and z are available in all the quantities. The highest common power of x is 2, the highest common power of y is 2, and the greatest common power of z is 3.

Therefore, the required greatest common factor = 18x²y²z³

Example 3.

Find the G.C.F of 20abc, 22a²b²c², and 24a³b³c².

Solution:

20abc = 2 * 2 * 5 * a * b * c

22a²b²c² = 2 * 11 * a * a * b * b * c * c

24a³b³c² = 2 * 2 * 2 * 3 * a * a * a * b * b * b * c * c

From the resolved factors of the above three monomials, the common factors are 2, a, b, c

Therefore, the required H.C.F. = 2 * a * b * c = 2abc