How to factor the following?

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Q.1. m^4 + 3m^2 + 4
Q.2. y^4 + 2y^2 + 9
Q.3. (2x+y)^2 - z^2
Q.4. 4y^4 - 16y^2 + 9
Q.5. m^2 + 6m + 9 - 4n^2
Q.6. 16y^2 - a^2 - 6ab - 9b^2

Oct 20th, 2017

Q.1. m^4 + 3m^2 + 4

= (m^2)^2 + 3m^2 + 4

Now a = 1  b = 3  c= 4 , then

 m^2 = (-b ± √b^2 - 4ac)/2a

Now

= (-3 ± √(3)^2 - 4(1)(4))/2(1)

= (- 3 ± √9 - 16)/2

= (- 3 ± √-7)/2

m^2 = (- 3 ± 7i)/2

Hence   m = √(- 3 ± 7i)/2

Q.2. y^4 + 2y^2 + 9

(y^2)^2 + 2y^2 + 9

a = 1  b = 2  c = 9

y^2 = (-b ± √b^2 - 4ac)/2a

Now

= (-(2) ± √(2)^2 - 4(1)(9))/2(1)

= (-2 ± √4 - 36)/2

= (-2 ± √- 32)/2

= (-2 ± √- 8 * 4)/2

= (-2 ± 2√- 8)/2

= 2(-1 ± √-8)/2

= (-1 ± 8i)

y^2 = (-1 ± 8i)

Hence  y = √(-1 ± 8i)

Q.3. (2x+y)^2 - z^2

(2x+y)^2 - z^2

= (2x+y - z)(2x + y + z)                    As a^2 - b^2 = (a - b)(a + b)

Q.4. 4y^4 - 16y^2 + 9

4(y^2)^2 - 16y^2 + 9

a = 4  b = -16  c = 9

Now

y^2 = (-b ± √b^2 - 4ac)/2a

= (-(-16) ± √(-16)^2 - 4(4)(9))/2(4)

= (16 ± √256  - 144)/8

= (16 ± √112)/8

= (16 ± √16 * 7)/8

= (16 ± 4√7)/8

= 4(4 ± √7)/8

= (4 ± √7)/2

y ^2 = (4 ± √7)/2

Hence y = √(4 ± √7)/2

Q.5. m^2 + 6m + 9 - 4n^2

= m^2 + 3m + 3m + 9 - 4n^2

= m(m + 3) + 3(m + 3) - 4n^2

= (m + 3)(m + 3) - 4n^2

= (m + 3)^2 - 4n^2

= (m + 3)^2 - (2n)^2

= (m + 3 - 2n)(m + 3 + 2n)                         As   a^2 - b^2 = (a - b)(a + b)

Q.6. 16y^2 - a^2 - 6ab - 9b^2

= 16y^2 - a^2 - 2*3*ab - 9b^2

= 16y^2 - (a^2 + 2*3*ab + 9b^2)

= 16y^2 - ((a)^2 + 2*3*ab + (3b)^2)

= 16y^2 - (a + 3b)^2                                     As a^2 + 2ab + b^2 = (a + b)^2

= (4y)^2 - (a + 3b)^2

= (4y - a + 3b)(4y + a + 3b)                    As   a^2 - b^2 = (a - b)(a + b)

Sep 29th, 2014

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