Factoring the Algebraic Expressions
To find the Least Common Multiple (LCM) of the given expressions, we first need to factor each expression completely.
Expression 1: \(x^4+x^2y^2 + y^4\)
This expression can be factored by adding and subtracting \(x^2y^2\) to complete the square:
- \(x^4+x^2y^2 + y^4 = (x^4 + 2x^2y^2 + y^4) - x^2y^2\)
- \(= (x^2+y^2)^2 - (xy)^2\)
- Using the difference of squares formula (\(a^2 - b^2 = (a-b)(a+b)\)):
- \(= (x^2+y^2 - xy)(x^2+y^2 + xy)\)
- Rearranging the terms within the factors:
- \(= (x^2 - xy + y^2)(x^2 + xy + y^2)\)
Expression 2: \(x^3y+y^4\)
Factor out the common term \(y\):
- \(x^3y+y^4 = y(x^3+y^3)\)
- Using the sum of cubes formula (\(a^3+b^3 = (a+b)(a^2-ab+b^2)\)):
- \(= y(x+y)(x^2 - xy + y^2)\)
Expression 3: \(x^4y^2 - x^3 y^3\)
Factor out the common term \(x^3y^2\):
- \(x^4y^2 - x^3 y^3 = x^3y^2(x - y)\)
Finding the LCM
The LCM is found by taking the highest power of each unique factor present in the factorizations of the expressions.
The unique factors are:
- \(x^3\) (highest power from Expression 3)
- \(y^2\) (highest power from Expression 3)
- \((x-y)\) (from Expression 3)
- \((x+y)\) (from Expression 2)
- \((x^2 - xy + y^2)\) (present in Expressions 1 and 2, highest power is 1)
- \((x^2 + xy + y^2)\) (from Expression 1)
Multiplying these factors together gives the LCM:
LCM \(= x^3 \cdot y^2 \cdot (x-y) \cdot (x+y) \cdot (x^2 - xy + y^2) \cdot (x^2 + xy + y^2)\)
Simplifying the LCM Expression
We can simplify the product of factors using algebraic identities:
- Recall the difference of cubes: \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\).
- Recall the sum of cubes: \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\).
- Also, recall the difference of sixth powers: \(a^6 - b^6 = (a^3)^2 - (b^3)^2 = (a^3 - b^3)(a^3 + b^3)\).
- Substituting the factored forms of cubes:
- \(x^6 - y^6 = [(x-y)(x^2+xy+y^2)] \cdot [(x+y)(x^2-xy+y^2)]\)
- Rearranging the terms, we see that \((x-y)(x+y)(x^2-xy+y^2)(x^2+xy+y^2)\) is exactly the product of the polynomial factors we found.
Therefore, the LCM can be written as:
LCM \(= x^3 y^2 (x^6 - y^6)\)
Conclusion
By factoring each polynomial and identifying the highest power of all unique factors, we determined the LCM. The LCM of \(x^4+x^2y^2 + y^4\), \(x^3y+y^4\), and \(x^4y^2 - x^3 y^3\) is \(x^3y^2 (x^6 - y^6)\).
Similar Questions
- If HCF of 768 and \(x^6y^2\) is \(32xy\) for natural numbers \(x \ge 2, y \ge 2\), then what is the value of \((x + y)\)?
- What is the HCF of \(2^{36}-1\) and \(2^{45}-1\)?
The HCF of \(x\) and \(y\) is \(H\). Consider the following statements in respect of the HCF of \(p=\frac{x^3 + y^3}{x^2-xy+y^2}\) and \(q=\frac{x^3-y^3}{x^2+xy+y^2}\) :
I. The HCF of \(p\) and \(q\) can be \(H\).
II. The HCF of \(p\) and \(q\) can be \(2H\).
Which of the statements given above is/are correct?
- What is the HCF of \(x^3 + y^3 + 3xy-1\) and \((x + y)^2 - 1\)?
- A number N is such that when divided by 4, 6, 7 or 9, it leaves 3 as remainder. What is the smallest 4-digit number that satisfies this property?
- A plank of wood 4·25 m long and 3·4 m wide is to be cut into square pieces of equal size. How many square pieces of largest size can be cut from the plank, if no wastage is allowed ?
- What is the HCF of \(x^4 - 13x^2y^2 - 300y^4\), \(x^3 - 4x^2y - 4xy^2 - 5y^3\) and \(x^3 - 125y^3\)?
- Let \(N\) be a 5-digit number. When \(N\) is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of \(N\)?
Let \(N\) be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of he following is correct?
Let \(n\) be a natural number. The HCF of \(n, n + 10\) is 10. If the LCM is \(x\) (a 2-digit number), then how many values of \(x\) are possible?
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
