**Resolve into factors**

**(i) x**^{4 }**+ y**^{4}**– 7x**^{2}**y**^{2}

Solution:

= x^{4} + y^{4 }– 2x^{2} y^{2} – 2x^{2} y^{2} – 7x^{2} y^{2}

= x^{4} + y^{4} + 2x^{2} y^{2} – 9x^{2} y^{2}

= (x^{2} + y^{2})^{2} – (3xy)^{2}

= (x^{2} + y^{2} + 3xy) (x^{2} + y^{2} – 3xy)

**(ii) 4x**^{4}**+ 25y**^{4}**+ 10x**^{2}**y**^{2}

Solution:

Take a = √2x ;b = √5y

a^{4} + b^{4} + a^{2} b^{2} = (a^{2} + b^{2} + ab) (a^{2} + b^{2} – ab)

4x^{4} + 25y^{4} + 10x^{2}y^{2} = (√2 x)^{4} + (√5 y)^{4} + 2 x^{2 }5 y^{2}

= [( 2 x^{2}) + ( 5 y^{2}) + (√10 xy] [( 2 x^{2}) + ( 5 y^{2}) – (√10 xy]

= (2x^{2} +5y^{2} + 10 xy) (2x^{2} +5y^{2} – 10 xy)

**(iii)9a ^{4}+100b^{4}+30a^{2}b^{2}**

Solution:

Take a = √3a ;b = √10b

a^{4} + b^{4} + a^{2} b^{2} = (a^{2} + b^{2} + ab) (a^{2} + b^{2} – ab)

9a^{4}+100b^{4}+30a^{2}b^{2} = (√3a)^{4}+(√10b)^{4}+3a^{2}.10b^{2}

= (3a^{2}+10b^{2}+30a^{2}b^{2}) (3a^{2}+10b^{2}– 30a^{2}b^{2})

**(iv)81a4 +9a2b2+b4**

Solution:

Take a = 3a ;b = b

a^{4} + b^{4} + a^{2} b^{2} = (a^{2} + b^{2} + ab) (a^{2} + b^{2} – ab)

81a^{4} +9a^{2}b^{2}+b^{4 }= (3a)^{4}+(3a)^{2}(b)^{2}+b^{4}

= (9a^{2}+b^{2}+3ab)(9a^{2}+b^{2}-3ab)

**(v) x**^{4}**– 6x**^{2}**y**^{2}**+ y**^{4}

= x^{4} + y^{4} + 2x^{2}y^{2} – 2x^{2}y^{2} – 6x^{2}y^{2}

= (x^{2} + y^{2})^{2} – ( 8 xy)^{2}

= (x^{2} + y^{2} + 8 xy) (x^{2} + y^{2} – 8 xy)

**vi) m**^{4}**+ n**^{4}**– 18m**^{2}**n**^{2}

= m^{4} + n^{4} + 2m^{2}n^{2} – 2m^{2}n^{2} – 18m^{2}n^{2}

= (m^{2} + n^{2})^{2} – 20 m^{2}n^{2}

= (m^{2} + n^{2} + √20 mn) (m^{2} + n^{2} – √20 mn)

**(vii) 4m**^{4}**+ 9n**^{4}**– 24m**^{2}**n**^{2}

= 4m^{4} + 9n^{4 }+ 12 m^{2}n^{2} – 12 m^{2}n^{2} – 24m^{2}n^{2}

= (2m^{2} + 3n^{2})^{2} – 36mn

= (2m^{2} + 3n^{2} + 6mn) (2m^{2} + 3n^{2} – 6mn)

**(viii) 9x**^{4 }**+ 4y**^{4}**+ 11x**^{2}**y**^{2 }

= 9x^{4 }+ 4y^{4} + 12x^{2}y^{2} – 12x^{2}y^{2} + 11x^{2}y^{2}

= (3x^{2})^{2} + (2y^{2})^{2} + 2 (3x)^{2} (2y)^{2} – x^{2}y^{2}

= (3x^{2} + 2y^{2})^{2} – x^{2}y^{2}

= (3x^{2} + 2y^{2 }+ xy) (3x^{2} + 2y^{2} – xy)

**Find the factors of the following**

**(i) x ^{4} + 9x^{2}+81**

Solution:

= x^{4}+9x^{2}+81+9x^{2}-9x^{2}

= (x^{2})^{2}+18x^{2}+81-9a^{2}

=(x^{2}+9)^{2}-(3a)^{2}

=( x^{2}+9 +3a)( x^{2}+9 – 3a)

**(ii) a**^{4}**+ 4a**^{2}**+ 16 **

Solution:

= a^{4} + 4a^{2} + 16 + 4a^{2 }– 4a^{2}

= (a^{2})^{2} + 8a^{2} + (4)^{2} – 4a^{2}

= (a^{2} + 4)^{2} – (2a)^{2}

= (a^{2} + 4 – 2a) (a^{2} + 4 + 2a)

**Factorise the following**

**(i) 64a**^{4}**+ 1 **

Solution:

Adding and subtracting 16a^{2} we get

64a4 + 1 + 16a^{2} – 16a^{2}

= (8a^{2})^{2} + 1 + 16a^{2} – 16a^{2}

= (8a^{2} + 1)^{2} – (4a)^{2}

= (8a^{2} + 1 + 4a) (8a^{2} + 1 – 4a)

**(ii) 3x**^{4}**+ 12y**^{4}

Solution:

3(x^{4} + 4y^{4})

3[(x^{2})^{2} + (^{2}y^{2})^{2}]

By adding and subtracting 2ab i. e.

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

= 3 [(x^{2})^{2} + 2(y^{2})^{2} + 4x^{2}y^{2} – 4x^{2}y^{2}]

= 3 [(x^{2} + 2y^{2})^{2} – (2xy)^{2}]

= 3 [(x^{2} + 2y2 + 2xy) (x^{2} + 2y^{2} – 2xy)]

**(iii) 4x**^{4}**+ 81y**^{4 }

Solution:

(2x^{2})^{2} + (9y^{2})^{2}

By adding and subtracting 2ab i. e.

2 x x^{2} x 9y^{2} = 36x^{2}y^{2 }

= (2x^{2})^{2} + (9y^{2})^{2} + 36x^{2}y^{2} – 36x^{2}y^{2 }

= (2x^{2} + 9y^{2})^{2} – (6xy)^{2}

= (2x^{2} + 9y^{2} + 6xy) (2x^{2} + 9y^{2} – 6xy)

**(iv) a**^{8}**– 16b**^{8}

Solution:

(a^{4})^{2} – (4b^{4})^{2}

Using a^{2} – b^{2} = (a + b) (a – b)

= (a^{4} + 4b^{4}) (a^{4} – 4b^{4})

= (a^{4} + 4b^{4}) [(a^{2})^{2} – (2b^{2})^{2}]

= (a^{4} + 4b^{4}) (a^{2} + 2b^{2}) (a^{2} – 2b^{2})

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