Math: factoring difference of cubes

The math tutor continues with factoring cubes.  Tutoring calculus, this formula is another standby.

Let’s turn to factoring x^3 – 64. Realizing that 64=4^3, you actually are facing x^3-4^3

You need the formula

a^3 – b^3=(a-b)(a^2 + ab + b^2)

For x^3-64, we substitute x for a and 4 for b. We arrive at

x^3-64=(x-4)(x^2+x(4)+16)=(x-4)(x^2+4x+16)

What about factoring 343x^6-8y^3? Since 343=7^3 and 8=2^3, we can rewrite the expression as
(7x^2)^3-(2y)^3

Carefully substituting 7x^2 for a and 2y for b, we arrive at

343x^6-8y^3=(7x^2-2y)((7x^2)^2+(7x^2)(2y)+(2y)^2))

Finally we simplify to

(7x^2-2y)(49x^4+14x^2y+4y^2)

The key with using these formulas is knowing how to rewrite your own expression so the proper substitution can be made.

Cheers,

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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