The L. C. M. of x2 - y2, x3 - y3 and x3 - x2y - xy2 + y3 is:
(x + y) (x - y)2 (x2 + y2 + xy)
Finding the Polynomial LCM
To find the Least Common Multiple (L. C. M.) of the given algebraic expressions, we need to factorize each expression completely. The L. C. M. is the product of the highest powers of all unique factors present in the factorized forms of the expressions. This process involves factorization and understanding algebraic expressions.
Factorizing the Algebraic Expressions
Let's factorize each polynomial step by step:
- Expression 1: \(x^2 - y^2\)
This is a difference of squares, which factors as:
\(x^2 - y^2 = (x - y)(x + y)\)
- Expression 2: \(x^3 - y^3\)
This is a difference of cubes, which factors as:
\(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)
- Expression 3: \(x^3 - x^2y - xy^2 + y^3\)
We can factor this by grouping terms:
\(x^3 - x^2y - xy^2 + y^3\)
Group the first two and last two terms:
\((x^3 - x^2y) - (xy^2 - y^3)\)
Factor out common terms from each group:
\(x^2(x - y) - y^2(x - y)\)
Now, factor out the common binomial factor \((x - y)\):
\((x - y)(x^2 - y^2)\)
The term \((x^2 - y^2)\) is a difference of squares, so we factor it further:
\((x - y)(x - y)(x + y)\)
Combine the identical factors:
\((x - y)^2(x + y)\)
Identifying Factors and Their Highest Powers
Now, let's list the factorized forms and identify all unique factors and their highest powers:
- \(x^2 - y^2 = (x - y)^1 (x + y)^1\)
- \(x^3 - y^3 = (x - y)^1 (x^2 + xy + y^2)^1\)
- \(x^3 - x^2y - xy^2 + y^3 = (x - y)^2 (x + y)^1\)
| Factor | Expression 1 Power | Expression 2 Power | Expression 3 Power | Highest Power (for LCM) |
|---|---|---|---|---|
| \((x - y)\) | 1 | 1 | 2 | 2 |
| \((x + y)\) | 1 | 0 | 1 | 1 |
| \((x^2 + xy + y^2)\) | 0 | 1 | 0 | 1 |
Calculating the Polynomial LCM
The Polynomial LCM is the product of the highest powers of all unique factors. Based on the table, the unique factors and their highest powers are:
- \((x - y)^2\)
- \((x + y)^1\)
- \((x^2 + xy + y^2)^1\)
Therefore, the Polynomial LCM is:
\(\text{L. C. M.} = (x - y)^2 \times (x + y)^1 \times (x^2 + xy + y^2)^1\)
\(\text{L. C. M.} = (x + y)(x - y)^2(x^2 + xy + y^2)\)
This calculation is crucial in finding the correct Polynomial LCM for these algebraic expressions using factorization techniques common in algebra.
The resulting expression is the Least Common Multiple of the given polynomials.
Important Questions from Evaluation of Limits
If $log 2 = 0.3010$ and $log 3 = 0.4771$, then the value of $log 36$ is
The series \(1 + \frac{2}{3} + {\left( {\frac{2}{3}} \right)^2} + ... + {\left( {\frac{2}{3}} \right)^{n - 1}}\) is:
The series \(\sum {\left( {\frac{1}{{np}}} \right)} \) is divergent if
If \(x + \frac{1}{x} = \sqrt{3}\), then the value of x18 + x12 + x6 + 1 is
If a + b + c = 5 and ab + bc + ca = 10, then the value of a3 + b3 + c3 - 3abc is
