Consider the following for the next items that follow: Let p = x4 - y2z2, q = y4 - z2x2, r = z4 - x2y2.
What is px2 + qy2 + rz2 equal to ?
(x2 + y2 + z2) (p + q + r)
Evaluating Algebraic Expressions with Given Terms
The problem asks us to evaluate the expression \(px^2 + qy^2 + rz^2\), where \(p\), \(q\), and \(r\) are defined in terms of \(x\), \(y\), and \(z\). We are given the following definitions:
- \(p = x^4 - y^2z^2\)
- \(q = y^4 - z^2x^2\)
- \(r = z^4 - x^2y^2\)
We need to find the value of the expression \(px^2 + qy^2 + rz^2\). To do this, we will substitute the given expressions for \(p\), \(q\), and \(r\) into the expression and simplify.
Step 1: Substitute p, q, and r into the expression
Substitute the expressions for \(p\), \(q\), and \(r\) into \(px^2 + qy^2 + rz^2\):
\(px^2 = (x^4 - y^2z^2)x^2 = x^4 \cdot x^2 - y^2z^2 \cdot x^2 = x^{4+2} - x^2y^2z^2 = x^6 - x^2y^2z^2\)
\(qy^2 = (y^4 - z^2x^2)y^2 = y^4 \cdot y^2 - z^2x^2 \cdot y^2 = y^{4+2} - x^2y^2z^2 = y^6 - x^2y^2z^2\)
\(rz^2 = (z^4 - x^2y^2)z^2 = z^4 \cdot z^2 - x^2y^2 \cdot z^2 = z^{4+2} - x^2y^2z^2 = z^6 - x^2y^2z^2\)
Step 2: Add the substituted terms
Now, add the results from Step 1:
\(px^2 + qy^2 + rz^2 = (x^6 - x^2y^2z^2) + (y^6 - x^2y^2z^2) + (z^6 - x^2y^2z^2)\)
Combine the terms:
\(px^2 + qy^2 + rz^2 = x^6 + y^6 + z^6 - x^2y^2z^2 - x^2y^2z^2 - x^2y^2z^2\)
\(px^2 + qy^2 + rz^2 = x^6 + y^6 + z^6 - 3x^2y^2z^2\)
So, the expression \(px^2 + qy^2 + rz^2\) simplifies to \(x^6 + y^6 + z^6 - 3x^2y^2z^2\). Now let's examine the given options.
Step 3: Evaluate the expressions in the options
Let's evaluate the first option: \((x^2 + y^2 + z^2)(p + q + r)\).
First, find \(p + q + r\):
\(p + q + r = (x^4 - y^2z^2) + (y^4 - z^2x^2) + (z^4 - x^2y^2)\)
\(p + q + r = x^4 + y^4 + z^4 - x^2y^2 - y^2z^2 - z^2x^2\)
Now, multiply \((x^2 + y^2 + z^2)\) by \((p + q + r)\):
\((x^2 + y^2 + z^2)(x^4 + y^4 + z^4 - x^2y^2 - y^2z^2 - z^2x^2)\)
Recall the algebraic identity: \(a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab - bc - ca)\).
Let \(a = x^2\), \(b = y^2\), and \(c = z^2\). Then \(a^2 = (x^2)^2 = x^4\), \(b^2 = (y^2)^2 = y^4\), \(c^2 = (z^2)^2 = z^4\). Also, \(ab = x^2y^2\), \(bc = y^2z^2\), and \(ca = z^2x^2\). And \(abc = x^2y^2z^2\).
Substituting these into the identity:
\((x^2)^3 + (y^2)^3 + (z^2)^3 - 3(x^2)(y^2)(z^2) = (x^2+y^2+z^2)((x^2)^2+(y^2)^2+(z^2)^2 - x^2y^2 - y^2z^2 - z^2x^2)\)
\(x^6 + y^6 + z^6 - 3x^2y^2z^2 = (x^2+y^2+z^2)(x^4+y^4+z^4 - x^2y^2 - y^2z^2 - z^2x^2)\)
Comparing this result with our calculation for \(p+q+r\), we see that \((p+q+r) = (x^4+y^4+z^4 - x^2y^2 - y^2z^2 - z^2x^2)\).
Therefore, \((x^2 + y^2 + z^2)(p + q + r) = (x^2 + y^2 + z^2)(x^4 + y^4 + z^4 - x^2y^2 - y^2z^2 - z^2x^2) = x^6 + y^6 + z^6 - 3x^2y^2z^2\).
Step 4: Compare the results
We found that \(px^2 + qy^2 + rz^2 = x^6 + y^6 + z^6 - 3x^2y^2z^2\).
We also found that \((x^2 + y^2 + z^2)(p + q + r) = x^6 + y^6 + z^6 - 3x^2y^2z^2\).
Since both expressions are equal to \(x^6 + y^6 + z^6 - 3x^2y^2z^2\), they are equal to each other.
\(px^2 + qy^2 + rz^2 = (x^2 + y^2 + z^2)(p + q + r)\).
This matches the first option.
| Expression | Simplified Form |
|---|---|
| \(px^2 + qy^2 + rz^2\) | \(x^6 + y^6 + z^6 - 3x^2y^2z^2\) |
| \((x^2 + y^2 + z^2)(p + q + r)\) | \(x^6 + y^6 + z^6 - 3x^2y^2z^2\) |
Revision Table: Key Algebraic Concepts
Understanding basic algebraic operations and identities is crucial for solving such problems. Here’s a quick review:
- Substitution: Replacing variables with their given expressions.
- Expansion: Multiplying out terms in brackets, e.g., \((a+b)(c+d) = ac+ad+bc+bd\).
- Combining Like Terms: Adding or subtracting terms that have the same variables raised to the same powers, e.g., \(2xy^2 - 5xy^2 = -3xy^2\).
- Algebraic Identities: Equations that are true for all values of the variables, e.g., \(a^2 - b^2 = (a-b)(a+b)\) or \(a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab - bc - ca)\).
Additional Information: Applications of Algebraic Identities
Algebraic identities are powerful tools used in various areas:
- Factoring Polynomials: Identities help in breaking down complex polynomial expressions into simpler factors.
- Simplifying Expressions: They provide shortcuts for simplifying expressions and making calculations easier, like the identity used in this problem.
- Solving Equations: Identities can transform equations into forms that are easier to solve.
- Calculus: Used for simplifying functions before differentiation or integration.
- Physics and Engineering: Identities appear in formulas and derivations in many scientific fields.
Mastering these fundamental skills and identities will significantly improve your ability to handle algebraic problems efficiently.
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