LCM of Polynomials By Factorization 20 Examples [ Best Que & Ans ]

LCM of Polynomials By Factorization | Least Common Multiply of Polynomials By Factorization | LCM Questions and Answers | LCM of Polynomials by middle term split

LCM of Polynomials By Factorization Example Questions And Answers

Que. 1 Find the L.C.M. of x²y, y²z, and z²x.

Ans. For finding out the LCM of x²y, y²z, and z²x, we have to factorize them one by one.
Let’s see the factorization:

x²y = x*x*y
y²z = y*y*z
z²x = z*z*x

L.C.M. of x²y, y²z, and z²x is x*x*y*y*z*z =x²y²z².

Que. 2. Find the Least Common Multiply of 5x³y, 12x²y³ and 15x⁴.

Ans. For finding out the LCM of 5x³y, 12x²y^3, and 15x⁴ we have to factorize them one by one.
Let’s see the factorization:

5x³y = 5*xxxy
12x²y³ =2*2*3*xxyyy
15x⁴ =3*5*xxxx

L.C.M. of 5x³y, 12x²y^3, and 15x⁴ is 2*2*3*5*xxxxyyy =60x⁴y³.

Que. 3. Find the LCM of Polynomials (x² – y²) and (x+y)² by factorization method.

Ans. First of all we will do the factors of these polynomials (x² – y²) and (x+y)².

Factors of Polynomial (x² – y²) by (a²-b²) =(a+b)(a-b)
(x² – y²) = (x+y)(x-y)

Factors of Polynomial (x+y)² are as follow:
(x+y)² = (x+y)(x+y)

LCM of the polynomials (x² – y²) and (x+y)² is (x+y)²(x-y).

Que 4. LCM of Polynomials By Factorization (x²-4) and (x² – 5x + 6).

Ans. Let’s see the factors of these polynomials (x²-4) and (x² – 5x + 6)

Fators of the polynomial (x²-4) by (a²-b²) =(a+b)(a-b)
(x²-4) = x²-2²
= (x+2)(x-2)

Factor of the Polynomial (x² – 5x + 6) by middle term split method
(x² – 5x + 6) = x² – 2x -3x + 6
= x(x-2)-3(x-2)
=(x-2)(x-3)

The LCM of (x²-4) and (x² – 5x + 6) is (x-2)(x+2)(x-3).

Que. 5. Find the LCM of 2x²y, 3y²z, and 4zx² by factorization method

Ans:. For finding LCM of Polynomials By Factorization Method
Let’s do the factorization of 2x²y, 3y²z, and 4zx²

2x²y = 2xxy
3y²z =3yyz
4zx² =2*2*xxz

Hence the LCM is 12x²y²z².

Que. 6. Find the LCM of Polynomials By Factorization of (x³ + 27) and (x² – 3x + 9).

Solution: Do the factors of (x³ + 27) and (x² – 3x + 9).

The factors of (x³ + 27) by Formula (a³+b³) = (a+b)(a²+b²-ab)

(x³ + 27) = x³ + 3³
= (x+3)(x²+9-3x)

The factors of (x² – 3x + 9) by Middle Term Split Method

(x² – 3x + 9) = x² – 3x + 9

The LCM of Polynomials (x³ + 27) and (x² – 3x + 9) is (x+3)(x² – 3x + 9)
= x³ + 27 ans.

Que. 7. LCM of Polynomials By Factorization Method (x²+x-6) and (x²+3x -10).

First of all we will do the factors of these two polynomials x²+x-6 and x²+3x -10 by middle term split method.

x²+x-6 = x² + 3x – 2x – 6
= x(x+3) -2(x+3)
=(x-2)(x+3)

and

x²+3x -10 = x²+ 5x -2x -10
=x(x+5)-2(x+5)
=(x-2)(x+5)

The LCM of polynomials x²+x-6 and x²+3x -10 is as follow:

(x-2)(x+3)(x+5) ans.

Que. 8. Find the LCM of polynomials (x-3)²(x+4) and (x-3)(x+4)².

Solution: For finding the least common multiplication of these polynomials we will do their factors.

Factors of the polynomial (x-3)²(x+4) are as follow:

(x-3)²(x+4) = (x-3)(x-3)(x+4)

and

Factors of this (x-3)(x+4)² polynomial are as follow:

(x-3)(x+4)² =(x-3)(x+4)(x+4)

The L.C.M. of the polynomials is (x-3)(x-3)(x-4)(x-4) = (x-3)²(x+4)².

Que. 9. Find the L.C.M. of polynomials 2(x²-9) and (x+3) by factorization method.

Solution: let’s do the factors of these both of polynomials 2(x²-9) and (x+3).

The factors of polynomial 2(x²-9) are as follow:
In this polynomial, we will use (a² – b²) a square minus b square formula.

2(x²-9) = 2(x²-3²)
= 2(x+3)(x-3)

and

(x+3) = (x+3)

From the above factors we can find X + 3 is the common factor. Hence the LCM of these polynomials 2(x²-9) and (x+3) is 2(x-3)(x+3).

LCM is =2(x²-9).

Que. 10. Find the least common multiplication of the following polynomials (x²-9) and (x³ + 27) by the factorization method.

Solution: LCM of Polynomials By Factorization
for finding the LCM of these polynomials first of all let’s do the factorization.

The factors of the polynomial (x²-9) are as follow:
In this polynomial, we will use (a² – b²) a square minus b square formula.

(x²-9) = (x²-3²)
= (x+3)(x-3)

and

The factors of (x³ + 27) by Formula (a³+b³) = (a+b)(a²+b²-ab)

(x³ + 27) = x³ + 3³
= (x+3)(x²+9-3x)

From the factors of both polynomials we can find X + 3 is the highest common factor.

Hence the LCM of the polynomials (x²-9) and (x³ + 27) is (x+3)(x-3)(x²+9-3x)

Here we can see that (x+3)(x²+9-3x) is similar to the expression of the formula (a³+b³) = (a+b)(a²+b²-ab).
Show LCM of the polynomial is (x-3)(x³ + 27).

Que. 11. Find the least common multiple of these x² – 7x + 12 and (x-4)² polynomials by the factorization method.

Solution: first of all we will do factors of these polynomials x² – 7x + 12 and (x-4)² And then we find out the least common multiply or L.C.M.

The factors of the polynomial x2 – 7x + 12 by middle term split are as follow:

x2 – 7x + 12
= x² -3x -4x +12
= x(x-3)-4(x-3)
= (x-3)(x-4)

and

Factors of the polynomial (x-4)² are as follow:
(x-4)² = (x-4)(x-4)

From the factors of these polynomials we can find that (x – 4) is the highest common factor.

So for finding the least common multiple I will take this common.

LCM = (x-3)(x-4)(x-4)
=(x-3)(x-4)².

Que. 12. LCM of Polynomials 3(x²-y²) and 6(x³-y³) by Factorization Method.

Solution: LCM of Polynomials By Factorization

let’s do the factorization of these polynomials to find the least common multiply.

3(x²-y²) and 6(x³-y³). These polynomials looks like the standard formulas of (a²-b²) and (a³-b³).

So we will expand all these as per the standard formula
3(x²-y²) = 3(x+y)(x-y)
6(x³-y³) = 2*3(x-y)(x²+xy+y²)

From the factors of these both polynomials, it is clear that the highest common factor is 3(x-y)
Hence the L.C.M. is 6(x-y)(x+y)(x²+xy+y²)
= 6(x²-y²)(x²+xy+y²).

Que. 13. LCM of Polynomials By Factorization Method (x³ – 27) and (x² – 7x + 12).

Solution: first of all we will do the factors of these polynomials (x³ – 27) and (x² – 7x + 12)

Do the factors of (x³ – 27) and (x² – 7x + 12).

The factors of (x³ – 27) by Formula (a³ – b³) = (a-b)(a²+b²+ab)

(x³ – 27) = x³ – 3³
= (x-3)(x²+9+3x)

And,

The factors of the polynomial x2 – 7x + 12 by middle term split are as follow:

x2 – 7x + 12
= x² -3x -4x +12
= x(x-3)-4(x-3)
= (x-3)(x-4)

Here we can see that Highest common factor is (x-3). So we will take it common for finding the LCM.

L.C.M. of the polynomials (x³ – 27) and (x² – 7x + 12) is (x-3)(x-4)(x²+9+3x).

Que. 14. Find the LCM of the following polynomials (x3 -9x) and (x2 -2x -3) by the factorization method.

Solution: first of all we will do the factors of these polynomials step by step.

The factors of the following polynomial (x³ -9x) and (x² -2x -3) are as following:
(x³ -9x) =x(x²-9)
= x(x²-3²)
= x(x+3)(x-3)

and

(x² -2x -3)
= x² -3x +x -3
= x(x-3)+1(x-3)
= (x+1)(x-3)

Here we can see (x-3) is the common factor of these polynomials.
L.C.M. of the polynomials are as follow: x(x+1)(x+3)(x-3) =x(x+1)(x²-9).

Que. 15. Find the LCM of the polynomials (x²-x-2) and (x²+x-6).

Solution: do the factors of these polynomials (x²-x-2) and (x²+x-6) by middle term split method.

(x²-x-2) = x²-2x +x -2
= x(x-2)+1(x-2)
=(x+1)(x-2)

and

(x²+x-6) = x²+3x -2x -6
= x(x+3)-2(x+3)
=(x-2)(x+3)

Here (x-2) is the common factor of both polynomials.

The L.C.M. is equal to (x+1)(x-2)(x+3).

Que. 16. Find the LCM of the given polynomials: (x²-x-12), (x²-9) and (x²-8x+16).

Solution: LCM of Polynomials By Factorization Method
In this question we have three polynomials. We will find out the factors of these polynomials one by one.

(x²-x-12) = x²-4x+3x-12 by middle term split
= x(x-4)+3(x-4)
= (x+3)(x-4)

(x²-9) = (x²-9)
= (x²-3²)
= (x+3)(x-3)

(x²-8x+16) = x²-8x+16
= x² – 4x – 4x + 16
= x(x-4) -4(x-4)
=(x-4)(x-4)

LCM of the polynomials is (x-3)(x+3)(x-4)².

Also, Read:

Top 20 Examples HCF of Polynomials By Factorization Method