निम्नलिखित का मान ज्ञात कीजिए: (i) 125x3 + y3 (ii) (2x−1)3 - (2x−1)3

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here are the factorizations for the given expressions. i) We need to factorize 125x³ + y³. This expression is in the form of a³ + b³. We use the identity a³ + b³ = (a+b)(a² − ab + b²). Step 1: Identify a and b. We can write 125x³ as (5x)³. So, a = 5x. And y³ is already in the form b³, so b = y. Step 2: Substitute a and b into the identity. 125x³ + y³ = (5x + y)((5x)² − (5x)(y) + y²) Step 3: Simplify the terms. = (5x + y)(25x² − 5xy + y²) The factored form is: (5x + y)(25x² − 5xy + y²) ii) We need to factorize 8x³ − y³. This expression is in the form of a³ − b³. We use the identity a³ − b³ = (a−b)(a² + ab + b²). Step 1: Identify a and b. We can write 8x³ as (2x)³. So, a = 2x. And y³ is already in the form b³, so b = y. Step 2: Substitute a and b into the identity. 8x³ − y³ = (2x − y)((2x)² + (2x)(y) + y²) Step 3: Simplify the terms. = (2x − y)(4x² + 2xy + y²) The factored form is: (2x − y)(4x² + 2xy + y²) iii) We need to factorize (2x − 5y)³ − (2x − y)³. This expression is in the form of A³ − B³. We use the identity A³ − B³ = (A−B)(A² + AB + B²). Step 1: Identify A and B. Let A = (2x − 5y) and B = (2x − y). Step 2: Calculate A−B. A−B = (2x − 5y) − (2x − y) = 2x − 5y − 2x + y = −4y Step 3: Calculate A², B², and AB. A² = (2x − 5y)² = (2x)² − 2(2x)(5y) + (5y)² = 4x² − 20xy + 25y²; B² = (2x − y)² = (2x)² − 2(2x)(y) + (y)² = 4x² − 4xy + y²; AB = (2x − 5y)(2x − y) = 2x(2x − y) − 5y(2x − y) = 4x² − 2xy − 10xy + 5y² = 4x² − 12xy + 5y² Step 4: Calculate A² + AB + B². A² + AB + B² = (4x² − 20xy + 25y²) + (4x² − 12xy + 5y²) + (4x² − 4xy + y²) Combine like terms: = (4x² + 4x² + 4x²) + (−20xy − 12xy − 4xy) + (25y² + 5y² + y²); = 12x² − 36xy + 31y²; Step 5: Substitute the results from Step 2 and Step 4 into the identity A³ − B³ = (A−B)(A² + AB + B²). (2x − 5y)³ − (2x − y)³ = (−4y)(12x² − 36xy + 31y²) Step 6: Distribute −4y into the trinomial. = −4y · 12x² − 4y · (−36xy) − 4y · 31y²; = −48x²y + 144xy² − 124y³ The factored form is: −48x²y + 144xy² − 124y³

निम्नलिखित का मान ज्ञात कीजिए: (i) 125x3 + y3 (ii) (2x-1)3 - (2x-1)3

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Step-by-step solution

Part i· 3 steps

Identify $a$ and $b$ .

Substitute $a$ and $b$ into the identity.

Simplify the terms.

Part ii· 3 steps
$(2x - y)(4x^2 + 2xy + y^2)$

Identify $a$ and $b$ .

Substitute $a$ and $b$ into the identity.

Simplify the terms.

Part iii· 6 steps
$-48x^2y + 144xy^2 - 124y^3$

Identify $A$ and $B$ .

Calculate $A-B$ .

Calculate $A^2$ , $B^2$ , and $AB$ .

Calculate $A^2 + AB + B^2$ .

Substitute the results from Step 2 and Step 4 into the identity $A^3 - B^3 = (A-B)(A^2 + AB + B^2)$ .

Distribute $-4y$ into the trinomial.

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निम्नलिखित का मान ज्ञात कीजिए: (i) 125x3 + y3 (ii) (2x-1)3 - (2x-1)3

Answers

Step-by-step solution

Part i· 3 steps

Identify aaa and bbb.

Substitute aaa and bbb into the identity.

Simplify the terms.

Part ii· 3 steps(2x−y)(4x2+2xy+y2)(2x - y)(4x^2 + 2xy + y^2)(2x−y)(4x2+2xy+y2)

Identify aaa and bbb.

Substitute aaa and bbb into the identity.

Simplify the terms.

Part iii· 6 steps−48x2y+144xy2−124y3-48x^2y + 144xy^2 - 124y^3−48x2y+144xy2−124y3

Identify AAA and BBB.

Calculate A−BA-BA−B.

Calculate A2A^2A2, B2B^2B2, and ABABAB.

Calculate A2+AB+B2A^2 + AB + B^2A2+AB+B2.

Substitute the results from Step 2 and Step 4 into the identity A3−B3=(A−B)(A2+AB+B2)A^3 - B^3 = (A-B)(A^2 + AB + B^2)A3−B3=(A−B)(A2+AB+B2).

Distribute −4y-4y−4y into the trinomial.

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Identify $a$ and $b$ .

Substitute $a$ and $b$ into the identity.

Part ii· 3 steps
$(2x - y)(4x^2 + 2xy + y^2)$

Identify $a$ and $b$ .

Substitute $a$ and $b$ into the identity.

Part iii· 6 steps
$-48x^2y + 144xy^2 - 124y^3$

Identify $A$ and $B$ .

Calculate $A-B$ .

Calculate $A^2$ , $B^2$ , and $AB$ .

Calculate $A^2 + AB + B^2$ .

Substitute the results from Step 2 and Step 4 into the identity $A^3 - B^3 = (A-B)(A^2 + AB + B^2)$ .

Distribute $-4y$ into the trinomial.