###### Advertisements

###### Advertisements

Factorize x^{4} + x^{2} y^{2} + y^{4}

###### Advertisements

#### Solution

Adding x^{2} y^{2} and subtracting x^{2} y^{2} to the given equation.

= x^{4} + x^{2} y^{2} + y^{4} + x^{2} y^{2} - x^{2} y^{2}

= x^{4} + 2x^{2} y^{2} + y^{4} - x^{2} y^{2}

= (x^{2} )^{2} + 2 x x^{2} x y^{2} + ( y^{2} )^{2} - ( xy)^{2}

Using identity a^{2} + 2ab + b^{2} = (a + b)^{2}

= (x^{2} + y^{2} )^{2} - ( xy)^{2}

Using identity a^{2} - b^{2} = (a + b)(a - b)

= (x^{2} + y^{2} + xy)(x^{2} + y^{2} - xy )

∴ x^{4} + x^{2} y^{2} + y^{4} = (x^{2} + y^{2} + xy)(x^{2} + y^{2} - xy)

#### APPEARS IN

#### RELATED QUESTIONS

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

Numbers x and y both squared and added.

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

Product of numbers y and z subtracted from 10.

Factorize `[x^2 + 1/x^2] - 4[x + 1/x] + 6`

Factorize `x^2 - sqrt3x - 6`

Factorize `x^2 + 12/35 x + 1/35`

Factorize the following expressions:

10x^{4}y – 10xy^{4}

Factorize the following expressions:

a^{12} + b^{12}

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

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: 64x^{2} + 81y^{2} + 144xy, when x = 11 and \[y = \frac{4}{3}\]

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?

4x^{2} − 12x + 7

The expression (*a* − *b*)^{3} + (*b *− *c*)^{3} + (*c* −*a*)^{3} can be factorized as

The value of \[\frac{(2 . 3 )^3 - 0 . 027}{(2 . 3 )^2 + 0 . 69 + 0 . 09}\]

Mark the correct alternative in each of the following: The factors of a^{2} − 1 − 2x − x^{2} are

The expression x^{4} + 4 can be factorized as

**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: `(1/2 "a" + 1/2 "b") (1/2 "a" - 1/2 "b")`

Multiply: (3x - 5y + 2)(5x - 4y - 3)

Divide: - 16ab^{2}c by 6abc

Divide: 8x + 24 by 4

Divide: - 50 + 40p by 10p

Divide: n^{2} - 2n + 1 by n - 1

Divide: 35a^{3} + 3a^{2}b - 2ab^{2} by 5a - b

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

The area of a rectangular field is 25x2 + 20xy + 3y2 square unit. If its length is 5x + 3y unit, find its breadth, Hence find its perimeter.

Divide: 2m^{3}n^{5} by - mn

Write in the form of an algebraic expression:

Perimeter (P) of a rectangle is two times the sum of its length (l) and its breadth (b).

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 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 constant term of the expression 2y – 6 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

29x + 13y

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

x + y

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?

The figure shows the dimensions of a wall having a window and a door of a room. Write an algebraic expression for the area of the wall to be painted.

The total number of planets of Sun can be denoted by the variable n.

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.

Sonu and Raj have to collect different kinds of leaves for science project. They go to a park where Sonu collects 12 leaves and Raj collects x leaves. After some time Sonu loses 3 leaves and Raj collects 2x leaves. Write an algebraic expression to find the total number of leaves collected by both of them.

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?

In the given figure, the length of a green side is given by g and that of the red side is given by p.

Write an expression for the following pattern. Also write an expression if 100 such shapes are joined together.

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

Commutative property of addition

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

Associative property of addition

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

Associative property of multiplication