###### Advertisements

###### Advertisements

If 3x = a + b + c, then the value of (x − a)^{3} + (x −b)^{3} + (x − c)^{3} − 3(x − a) (x − b) (x −c) is

#### Options

a + b + c

(a − b) (b − c) (c − a)

0

none of these

###### Advertisements

#### Solution

The given expression is

(x − a)^{3} + (x −b)^{3} + (x − c)^{3} − 3(x − a) (x − b) (x −c)

Recall the formula

`a^3 +b^3 +c^3 - 3abc = (a+b+c)(a^2 + b^2+ c^2 - ab - bc +_ ca)`

Using the above formula the given expression becomes

(x − a)^{3} + (x −b)^{3} + (x − c)^{3} − 3(x − a) (x − b) (x −c)

`{(x-a) +(x-b) + (x-c)} {(x-a)^2+(x-b)^2+(x-c)^2 -(x-a)(x-b)-(x-b)(x-c) - (x-c)(x-a)}`

`(x-a +x-b+x -c) {(x-a)^2+(x-b)^2+(x-c)^2 -(x-a)(x-b)-(x-b)(x-c) - (x-c)(x-a)} `

`(3x-a -b-c) {(x-a)^2+(x-b)^2+(x-c)^2 -(x-a)(x-b)-(x-b)(x-c) - (x-c)(x-a)}`

Given that

`3x = (a+b+c)`

` ⇒ 3x -a-b-c =0`

Therefore the value of the given expression is

(x − a)^{3} + (x −b)^{3} + (x − c)^{3} − 3(x − a) (x − b) (x −c)

`= 0.{(x-a)^2+(x-b)^2+(x-c)^2 -(x-a)(x-b)-(x-b)(x-c) - (x-c)(x-a)}`

`=0`

#### APPEARS IN

#### RELATED QUESTIONS

Get the algebraic expression in the following case using variables, constants and arithmetic operation.

Subtraction of z from y

Factorize: a (a + b)^{3} - 3a^{2}b (a + b)

Factorize x( x - 2)( x - 4) + 4x - 8

Factorize `x^2 - 2sqrt2x - 30`

Given possible expressions for the length and breadth of the rectangle having 35y^{2} + 13y – 12 as its area.

Factorize the following expressions:

54x^{6}y + 2x^{3}y^{4}

Factorize the following expressions:

(a - 2b)^{3} - 512b^{3}

`1/27 x^3 - y^3 + 125z^3 + 5xyz`

`(x/2 + y + z/3)^3 + (x/2 + (2y)/3 + z)^3 + (-(5x)/6 - y/3 - (4z)/3)^3`

`2sqrt2a^3 + 3sqrt3b^3 + c^3 - 3 sqrt6abc`

If \[x^2 + \frac{1}{x^2} = 18,\] find the values of \[x + \frac{1}{x} \text { and } x - \frac{1}{x} .\]

If x + y = 4 and xy = 2, find the value of x^{2} + y^{2}

If 3x + 5y = 11 and xy = 2, find the value of 9x^{2}^{ }+ 25y^{2}

Find the value of the following expression: 16*x*^{2} + 24*x* + 9, when \[x = \frac{7}{4}\]

Find the value of the following expression: 81x^{2} + 16y^{2} − 72xy, when \[x = \frac{2}{3}\] and \[y = \frac{3}{4}\]

If x^{2} + y^{2} = 29 and xy = 2, find the value of *x + y.*

What must be added to the following expression to make it a whole square?

4*x*^{2}^{ }− 20*x* + 20

27*x*^{3} − *y*^{3} − *z*^{3} − 9*xyz*

**Separate the constants and variables from the following :**

`-7,7+"x",7"x"+"yz",sqrt5,sqrt("xy"),(3"yz")/8,4.5"y"-3"x",`

8 −5, 8 − 5x, 8x −5y × p and 3y^{2}z ÷ 4x

Write the number of the term of the following polynomial.

5x^{2} + 3 x ax

Write the number of the term of the following polynomial.

ax ÷ 4 – 7

Write the number of the term of the following polynomial.

ax – by + y x z

Write the number of the term of the following polynomial.

23 + a x b ÷ 2

**Separate monomials, binomials, trinomials and polynomials from the following algebraic expressions :**

8 − 3x, xy^{2}, 3y^{2} − 5y + 8, 9x − 3x^{2} + 15x^{3} − 7,

3x × 5y, 3x ÷ 5y, 2y ÷ 7 + 3x − 7 and 4 − ax^{2} + bx + y

Evaluate: (3c - 5d)(4c - 6d)

Evaluate: (a^{2} + 2ab + b^{2})(a + b)

Evaluate: (8 - 12x + 7x^{2} - 6x^{3})(5 - 2x)

Evaluate: (4x^{2} - 4x + 1)(2x^{3} - 3x^{2} + 2)

Multiply: (2x + 3y)(2x - 3y)

Divide: - 50 + 40p by 10p

Divide: 6x^{2} + 5x - 6 by 2x + 3

Divide: 9x^{2} - 24xy + 16y^{2} by 3x- 4y

Divide: 6x^{3} + 5x^{2} - 21x + 10 by 3x - 2

The area of a rectangle is 6x^{2} – 4xy – 10y^{2} square unit and its length is 2x + 2y unit. Find its breadth.

Write in the form of an algebraic expression:

Perimeter (P) of a square is four times its side.

Write in the form of an algebraic expression:

Area of a square is square of its side.

Write in the form of an algebraic expression:

Surface area of a cube is six times the square of its edge.

Express the following as an algebraic expression:

The sum of x and y minus m.

Express the following as an algebraic expression:

The product of x and y divided by m.

Express the following as an algebraic expression:

The subtraction of 5m from 3n and then adding 9p to it.

Express the following as an algebraic expression:

The product of 12, x, y and z minus the product of 5, m and n.

Express the following as an algebraic expression:

Sum of p and 2r – s minus sum of a and 3n + 4x.

**Construct a formula for the following:**

Total wages (₹ W) of a man whose basic wage is (₹ B) for t hours week plus (₹ R) per hour, if he Works a total of T hours.

Find the average (A) of four quantities p, q, r and s. If A = 6, p = 3, q = 5 and r = 7; find the value of s.

**Solve the following equation.**

4(x + 12) = 8

**Solve the following equation.**

3y + 4 = 5y - 6

**Choose the right answer from the option.**

The simplest form of 5 ÷ `(3/2) - 1/3` is _______.

**Choose the right answer from the option.**

The solution of the equation `"3x" - 1/2 = 5/2 + "x"` is _____.

**Choose the right answer from the option.**

Which of the following expressions has the value 37?

Write the coefficient of x^{2} and x in the following polynomials

`4 + 2/5x^2 - 3x`

Write the coefficient of x^{2} and x in the following polynomials

`6 - 2x^2 + 3x^3 - sqrt(7)x`

Write the coefficient of x^{2} and x in the following polynomials

πx^{2} – x + 2

Write the coefficient of x^{2} and x in the following polynomials

`sqrt(3)x^2 + sqrt(2)x + 0.5`

Write the coefficient of x^{2} and x in the following polynomials

`x^2 - 7/2 x + 8`

Complete the table.

× | 2x^{2} |
−2xy | x^{4}y^{3} |
2xyz | (___)xz^{2} |

x^{4} |
|||||

(___) | 4x^{5}y^{4} |
||||

−x^{2}y |
|||||

2y^{2}z |
−10xy^{2}z^{3} |
||||

−3xyz | |||||

(___) | −14xyz^{2} |

The largest number of the three consecutive numbers is x + 1, then the smallest number is ________

The variable in the expression 16x – 7 is __________

When y = –1, the value of the expression 2y – 1 is 3

Write the variables, constant and terms of the following expression

18 + x – y

Write the variables, constant and terms of the following expression

7p – 4q + 5

Write the variables, constant and terms of the following expression

b + 2

If x = 2 and y = 3, then find the value of the following expressions

2x – 3y

If x = 2 and y = 3, then find the value of the following expressions

4y – x

If x = 2 and y = 3, then find the value of the following expressions

x + 1 – y

An algebraic statement which is equivalent to the verbal statement “Three times the sum of ‘x’ and ‘y’ is

The value of 7a – 4b when a = 3, b = 2 is

Give an algebraic equation for the following statement:

“The difference between the area and perimeter of a rectangle is 20”.

3p^{2} – 5pq + 2q^{2} + 6pq – q^{2} + pq is a

In a polynomial, the exponents of the variables are always ______.

Which of the following is correct?

An equation is true for all values of its variables.

In the formula, area of circle = πr^{2}, the numerical constant of the expression πr^{2} is ______.

If Rohit has 5xy toffees and Shantanu has 20yx toffees, then Shantanu has ______ more toffees.

A taxi service charges ₹ 8 per km and levies a fixed charge of ₹ 50. Write an algebraic expression for the above situation, if the taxi is hired for x km.

Shiv works in a mall and gets paid ₹ 50 per hour. Last week he worked for 7 hours and this week he will work for x hours. Write an algebraic expression for the money paid to him for both the weeks.

A school has a rectangular playground with length x and breadth y and a square lawn with side x as shown in the figure given below. What is the total perimeter of both of them combined together?

Rohan's mother gave him ₹ 3xy^{2} and his father gave him ₹ 5(xy^{2} + 2). Out of this total money he spent ₹ (10 – 3xy^{2}) on his birthday party. How much money is left with him?

A triangle is made up of 2 red sticks and 1 blue sticks . The length of a red stick is given by r and that of a blue stick is given by b. Using this information, write an expression for the total length of sticks in the pattern given below:

Express the following properties with variables x, y and z.

Associative property of multiplication

Express the following properties with variables x, y and z.

Distributive property of multiplication over addition