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, 12x2y3, 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, 12x2y3, 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.
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