The value of $ (x - y)(x + y) + (y - z)(y + z) + (z - x) (z + ... | Mathematics MCQ | Examoogle

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The value of $(x - y)(x + y) + (y - z)(y + z) + (z - x) (z + x)$ is:

$x^2 + y^2 + z^2$

$x + y + z$

$0$

$xy + yz + zx$

Correct Answer

📐 MA • Math Direct FormulaMathematicsAlgebra

Option C

$0$

Quick Summary:

Use the algebraic identity $(a - b)(a + b) = a^2 - b^2$ for each term and sum them up.

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📐MAMath SolutionDirect Formula

📋 Given

Expression: $(x - y)(x + y) + (y - z)(y + z) + (z - x)(z + x)$

🔢 Formula Used

(a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)(a+b)=a2−b2

⚡ Exam Hall Shortcut / Speed Trick

Use the algebraic identity $(a - b)(a + b) = a^2 - b^2$ for each term and sum them up.

⚠️ Common Student Trap / Pitfall

Students might expand incorrectly or misapply signs when combining terms.

🔢 Step-by-Step Solution

Step 1

Apply the algebraic identity $(a - b)(a + b) = a^2 - b^2$ to the first term $(x - y)(x + y)$ .

(x−y)(x+y)=x2−y2(x - y)(x + y) = x^2 - y^2(x−y)(x+y)=x2−y2

Step 2

Apply the identity to the second and third terms.

(y−z)(y+z)=y2−z2,(z−x)(z+x)=z2−x2(y - z)(y + z) = y^2 - z^2, \quad (z - x)(z + x) = z^2 - x^2(y−z)(y+z)=y2−z2,(z−x)(z+x)=z2−x2

Step 3

Add all three expanded terms together.

(x2−y2)+(y2−z2)+(z2−x2)(x^2 - y^2) + (y^2 - z^2) + (z^2 - x^2)(x2−y2)+(y2−z2)+(z2−x2)

Step 4

Simplify by canceling out opposite terms: $x^2 - x^2 = 0$ , $-y^2 + y^2 = 0$ , and $-z^2 + z^2 = 0$ .

x2−x2+y2−y2+z2−z2=0x^2 - x^2 + y^2 - y^2 + z^2 - z^2 = 0x2−x2+y2−y2+z2−z2=0

✅

C is correct because the expression simplifies to $0$ after applying the difference of squares identity to each term.

Core Concepts Used

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Algebraic Identities Difference of Squares Polynomial Simplification

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Recognizing standard algebraic identities helps solve polynomial simplification problems in seconds across various competitive exams.

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