How to factor the difference a^4-b^16?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to factor a^4 - b^16

we use x^2 - y^2 = (x +y)(x - y)

a^4 - b^16

=> (a^2 - b^8)(a^2 + b^8)

=> (a - b^4)(a + b^4)(a^2 + b^8)

Therefore a^4 - b^16 = (a - b^4)(a + b^4)(a^2 + b^8)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll use the power property:

a^4 = a^2*2 = (a^2)^2

b^16 = b^2*8 = (b^8)^2

we'll write the terms in this way:

(a^2)^2 - (b^8)^2

We notice that we've get a difference of squares:

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

We'll put x = a^2 and y = b^8:

(a^2)^2 - (b^8)^2 = (a^2 - b^8)(a^2 + b^8) (1)

We notice that the factor a^2 - b^8 is also a difference of squares:

a^2 - b^8 = (a)^2 - (b^4)^2

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

We'll substitute (2) in (1):

a^4-b^16 = (a - b^4)(a + b^4)(a^2 + b^8)

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