If x + y = 1, then what is the value of x 3+ 3xy + y 3?
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Understanding the Problem: Evaluating an Algebraic Expression
The question asks us to find the value of the expression \(x^3 + 3xy + y^3\) given the condition that \(x + y = 1\).
We need to use the given condition to simplify or relate it to the expression we need to evaluate.
Connecting the Expression to Algebraic Identities
The expression \(x^3 + y^3 + 3xy\) looks similar to the expansion of a cubic term. Let's recall the algebraic identity for the cube of a sum:
For any two variables, say 'a' and 'b', the cube of their sum \((a+b)\) is given by:
\((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
This identity can also be written as:
\((a+b)^3 = a^3 + b^3 + 3ab(a+b)\)
Applying the Identity with the Given Condition
Let's apply the second form of the identity using 'x' and 'y':
\((x+y)^3 = x^3 + y^3 + 3xy(x+y)\)
We are given the condition \(x + y = 1\). We can substitute this value into the identity:
\((1)^3 = x^3 + y^3 + 3xy(1)\)
Simplifying to Find the Value
Now, let's simplify the equation:
\(1^3 = 1\)
And \(3xy(1) = 3xy\)
So, the equation becomes:
\(1 = x^3 + y^3 + 3xy\)
The expression we need to find the value of is \(x^3 + 3xy + y^3\). Comparing this with the equation we derived, we see that:
\(x^3 + y^3 + 3xy = 1\)
Therefore, the value of \(x^3 + 3xy + y^3\) is 1.
Step-by-Step Solution Summary
- Identify the given condition: \(x + y = 1\).
- Identify the expression to evaluate: \(x^3 + 3xy + y^3\).
- Recall the cubic identity: \((a+b)^3 = a^3 + b^3 + 3ab(a+b)\).
- Substitute 'a' with 'x' and 'b' with 'y': \((x+y)^3 = x^3 + y^3 + 3xy(x+y)\).
- Substitute the given condition \(x + y = 1\) into the identity: \((1)^3 = x^3 + y^3 + 3xy(1)\).
- Simplify the equation: \(1 = x^3 + y^3 + 3xy\).
- Rearrange the terms on the right side to match the expression: \(1 = x^3 + 3xy + y^3\).
- Conclude the value of the expression is 1.
Revision Table: Key Algebraic Identities
Understanding fundamental algebraic identities is crucial for solving such problems. Here is a table of some common identities:
| Identity | Formula |
|---|---|
| Square of a sum | \((a+b)^2 = a^2 + 2ab + b^2\) |
| Square of a difference | \((a-b)^2 = a^2 - 2ab + b^2\) |
| Difference of squares | \(a^2 - b^2 = (a-b)(a+b)\) |
| Sum of cubes | \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\) |
| Difference of cubes | \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\) |
| Cube of a sum | \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) or \(a^3 + b^3 + 3ab(a+b)\) |
| Cube of a difference | \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\) or \(a^3 - b^3 - 3ab(a-b)\) |
Additional Information: Why Identities are Useful
Algebraic identities are equations that are true for all possible values of the variables involved. They are powerful tools in algebra because they allow us to simplify expressions, factor polynomials, and solve equations more easily.
- Identities provide shortcuts for expanding or factoring expressions without performing lengthy multiplications.
- They help in transforming expressions into equivalent forms that might be easier to work with in a specific context.
- Recognizing patterns that match algebraic identities is a key skill in algebraic manipulation.
In this specific problem, recognizing that \(x^3 + y^3 + 3xy\) is part of the expansion of \((x+y)^3\) when \(x+y\) is known was the key to finding the solution efficiently.
Important Questions from Identities
The coefficient of y in the expansion of (2y – 5) 3, is:
If x + y = 2 and \(\frac{1}{x}+\frac{1}{y}=\frac{18}{5}\) , then the value of (x 3+ y 3) is:
If x - y = 11 and \(\rm \frac{1}{x} - \frac{1}{y} = \frac{11}{24}\) then the value of x 3 - y 3 + x 2y 2 ?
If 2x 2- 8x - 1 = 0, then what is the value of \(\rm 8x^3 - \frac{1}{x^3}\) ?
If \(\rm x+ \frac{1}{x} = 4,\) then the value of \(\rm x^5 + \frac{1}{x^5}\) is:
