Using multiplication properties calculate (a-2b)(a^2+4b^2)(a+2b)

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We need to simplify: (a-2b)(a^2+4b^2)(a+2b)

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

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

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

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

=> a^4 - 16b^4

The required result is a^4 - 16b^4

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, we'll apply the commutative property for the 2nd and 3rd factors:

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

We'll re-write the product:

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

We notice that the product of the first 2 factors could be replaced by the difference of squares:

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

This product could be also replaced by its equivalent difference of squares:

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

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

The result of the product is: (a-2b)(a^2+4b^2)(a+2b)=a^4 - 16b^4

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