**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 x^{2}y, y^{2}z, and z^{2}x.

**Ans.** For finding out the LCM of x^{2}y, y^{2}z, and z^{2}x, we have to factorize them one by one.

Let’s see the factorization:

x^{2}y = x*x*y

y^{2}z = y*y*z

z^{2}x = z*z*x

L.C.M. of x^{2}y, y^{2}z, and z^{2}x is x*x*y*y*z*z =x^{2}y^{2}z^{2}.

#### Que. 2. Find the Least Common Multiply of 5x^{3}y, 12x^{2}y^{3} and 15x^{4}.

Ans. For finding out the LCM of 5x^{3}y, 12x^{2}y^{3,} and 15x^{4} we have to factorize them one by one.

Let’s see the factorization:

5x^{3}y = 5*xxxy

12x^{2}y^{3} =2*2*3*xxyyy

15x^{4} =3*5*xxxx

L.C.M. of 5x^{3}y, 12x^{2}y^{3,} and 15x^{4} is 2*2*3*5*xxxxyyy =60x^{4}y^{3}.

#### Que. 3. Find the LCM of Polynomials (x^{2} – y^{2}) and (x+y)^{2} by factorization method.

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

Factors of Polynomial (x^{2} – y^{2}) by (a^{2}-b^{2}) =(a+b)(a-b)

(x^{2} – y^{2}) = (x+y)(x-y)

Factors of Polynomial (x+y)^{2} are as follow:

(x+y)^{2} = (x+y)(x+y)

LCM of the polynomials (x^{2} – y^{2}) and (x+y)^{2} is (x+y)^{2}(x-y).

#### Que 4. LCM of Polynomials By Factorization (x^{2}-4) and (x^{2} – 5x + 6).

Ans. Let’s see the factors of these polynomials (x^{2}-4) and (x^{2} – 5x + 6)

Fators of the polynomial (x^{2}-4) by (a^{2}-b^{2}) =(a+b)(a-b)

(x^{2}-4) = x^{2}-2^{2}

= (x+2)(x-2)

Factor of the Polynomial (x^{2} – 5x + 6) by middle term split method

(x^{2} – 5x + 6) = x^{2} – 2x -3x + 6

= x(x-2)-3(x-2)

=(x-2)(x-3)

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

#### Que. 5. Find the LCM of 2x^{2}y, 3y^{2}z, and 4zx^{2} by factorization method

Ans:. For finding LCM of Polynomials By Factorization Method

Let’s do the factorization of 2x^{2}y, 3y^{2}z, and 4zx^{2}

2x^{2}y = 2xxy

3y^{2}z =3yyz

4zx^{2} =2*2*xxz

Hence the LCM is 12x^{2}y^{2}z^{2}.

#### Que. 6. Find the LCM of Polynomials By Factorization of (x^{3} + 27) and (x^{2} – 3x + 9).

Solution: Do the factors of (x^{3} + 27) and (x^{2} – 3x + 9).

The factors of (x^{3} + 27) by Formula (a^{3}+b^{3}) = (a+b)(a^{2}+b^{2}-ab)

(x^{3} + 27) = x^{3} + 3^{3}

= (x+3)(x^{2}+9-3x)

The factors of (x^{2} – 3x + 9) by Middle Term Split Method

(x^{2} – 3x + 9) = x^{2} – 3x + 9

The LCM of Polynomials (x^{3} + 27) and (x^{2} – 3x + 9) is (x+3)(x^{2} – 3x + 9)

= x^{3} + 27 ans.

#### Que. 7. LCM of Polynomials By Factorization Method (x^{2}+x-6) and (x^{2}+3x -10).

First of all we will do the factors of these two polynomials x^{2}+x-6 and x^{2}+3x -10 by middle term split method.

x^{2}+x-6 = x^{2} + 3x – 2x – 6

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

=(x-2)(x+3)

and

x^{2}+3x -10 = x^{2}+ 5x -2x -10

=x(x+5)-2(x+5)

=(x-2)(x+5)

The LCM of polynomials x^{2}+x-6 and x^{2}+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(x^{2}-9) and (x+3) by factorization method.

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

The factors of polynomial 2(x^{2}-9) are as follow:

In this polynomial, we will use (a^{2} – b^{2}) a square minus b square formula.

2(x^{2}-9) = 2(x^{2}-3^{2})

= 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(x^{2}-9) and (x+3) is 2(x-3)(x+3).

LCM is =2(x^{2}-9).

#### Que. 10. Find the least common multiplication of the following polynomials (x^{2}-9) and (x^{3} + 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 (x^{2}-9) are as follow:

In this polynomial, we will use (a^{2} – b^{2}) a square minus b square formula.

(x^{2}-9) = (x^{2}-3^{2})

= (x+3)(x-3)

and

The factors of (x^{3} + 27) by Formula (a^{3}+b^{3}) = (a+b)(a^{2}+b^{2}-ab)

(x^{3} + 27) = x^{3} + 3^{3}

= (x+3)(x^{2}+9-3x)

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

Hence the LCM of the polynomials (x^{2}-9) and (x^{3} + 27) is (x+3)(x-3)(x^{2}+9-3x)

Here we can see that (x+3)(x^{2}+9-3x) is similar to the expression of the formula (a^{3}+b^{3}) = (a+b)(a^{2}+b^{2}-ab).

Show LCM of the polynomial is (x-3)(x^{3} + 27).

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

Solution: first of all we will do factors of these polynomials x^{2} – 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

= x^{2} -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(x^{2}-y^{2}) and 6(x^{3}-y^{3}) by Factorization Method.

Solution: LCM of Polynomials By Factorization

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

3(x^{2}-y^{2}) and 6(x^{3}-y^{3}). These polynomials looks like the standard formulas of (a^{2}-b^{2}) and (a^{3}-b^{3}).

So we will expand all these as per the standard formula

3(x^{2}-y^{2}) = 3(x+y)(x-y)

6(x^{3}-y^{3}) = 2*3(x-y)(x^{2}+xy+y^{2})

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)(x^{2}+xy+y^{2})

= 6(x^{2}-y^{2})(x^{2}+xy+y^{2}).

#### Que. 13. LCM of Polynomials By Factorization Method (x^{3} – 27) and (x^{2} – 7x + 12).

Solution: first of all we will do the factors of these polynomials (x^{3} – 27) and (x^{2} – 7x + 12)

Do the factors of (x^{3} – 27) and (x^{2} – 7x + 12).

The factors of (x^{3} – 27) by Formula (a^{3} – b^{3}) = (a-b)(a^{2}+b^{2}+ab)

(x^{3} – 27) = x^{3} – 3^{3}

= (x-3)(x^{2}+9+3x)

And,

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

x2 – 7x + 12

= x^{2} -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 (x^{3} – 27) and (x^{2} – 7x + 12) is (x-3)(x-4)(x^{2}+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 (x^{3} -9x) and (x^{2} -2x -3) are as following:

(x^{3} -9x) =x(x^{2}-9)

= x(x^{2}-3^{2})

= x(x+3)(x-3)

and

(x^{2} -2x -3)

= x^{2} -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)(x^{2}-9).

#### Que. 15. Find the LCM of the polynomials (x^{2}-x-2) and (x^{2}+x-6).

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

(x^{2}-x-2) = x^{2}-2x +x -2

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

=(x+1)(x-2)

and

(x^{2}+x-6) = x^{2}+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: (x^{2}-x-12), (x^{2}-9) and (x^{2}-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.

(x^{2}-x-12) = x^{2}-4x+3x-12 by middle term split

= x(x-4)+3(x-4)

= (x+3)(x-4)

(x^{2}-9) = (x^{2}-9)

= (x^{2}-3^{2})

= (x+3)(x-3)

(x^{2}-8x+16) = x^{2}-8x+16

= x^{2} – 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}.

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