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Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]?
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Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]?
Verification of the given equation:
1. Let's start by expanding the right side of the equation:
\[
\begin{aligned}
&(x+y+z)[(x - y)^2+(y-z)^2+(z-x)^2] \\
&= (x+y+z)[x^2-2xy+y^2+y^2-2yz+z^2+z^2-2zx+x^2] \\
&= (x+y+z)(x^2+y^2+z^2-2(xy+yz+zx))
\end{aligned}
\]
2. Now, let's expand the left side of the equation:
\[
\begin{aligned}
&x^3+y+z^3-3xyz \\
&= x^3+z^3+y-3xyz \\
&= x^3+z^3+y-3xyz
\end{aligned}
\]
3. Simplifying the left side further:
\[
\begin{aligned}
&x^3+y+z^3-3xyz \\
&= x^3+z^3+(x+y+z-3xyz) \\
&= (x+y+z)(x^2+y^2+z^2-xy-yz-zx)
\end{aligned}
\]
4. Comparing the left side and the right side of the equation, we see that both sides are equal:
\[
(x+y+z)(x^2+y^2+z^2-xy-yz-zx) = (x+y+z)(x^2+y^2+z^2-2(xy+yz+zx))
\]
5. Hence, the given equation is verified to be true.
1. Let's start by expanding the right side of the equation:
\[
\begin{aligned}
&(x+y+z)[(x - y)^2+(y-z)^2+(z-x)^2] \\
&= (x+y+z)[x^2-2xy+y^2+y^2-2yz+z^2+z^2-2zx+x^2] \\
&= (x+y+z)(x^2+y^2+z^2-2(xy+yz+zx))
\end{aligned}
\]
2. Now, let's expand the left side of the equation:
\[
\begin{aligned}
&x^3+y+z^3-3xyz \\
&= x^3+z^3+y-3xyz \\
&= x^3+z^3+y-3xyz
\end{aligned}
\]
3. Simplifying the left side further:
\[
\begin{aligned}
&x^3+y+z^3-3xyz \\
&= x^3+z^3+(x+y+z-3xyz) \\
&= (x+y+z)(x^2+y^2+z^2-xy-yz-zx)
\end{aligned}
\]
4. Comparing the left side and the right side of the equation, we see that both sides are equal:
\[
(x+y+z)(x^2+y^2+z^2-xy-yz-zx) = (x+y+z)(x^2+y^2+z^2-2(xy+yz+zx))
\]
5. Hence, the given equation is verified to be true.
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Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]?
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Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]?.
Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Verify that: x³+y+z³-3xyz=(x+y+z)[(x - y)²+(y-2)²+(z-x)²]?.
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