Question

Which products result in a sum or difference of cubes? Check all that apply. (x – 4)(x2 4x – 16) (x – 1)(x2 – x 1) (x – 1)(x2 x 1) (x 1)( x – 1) (x 4)(x2 – 4x 16) (x 4)(x2 4x 16).

Answer 1

You can open those brackets and distribute one brackets content with multiplication to other bracket contents.

The products which result in a sum or difference of cubes is given by:

Option D: $(x+4)(x^2 -4x + 16)$

What is sum or difference of cubes?

If there are two terms which are with power 3 (ie cubed), and they both are in addition or subtraction, then that is called sum or difference of cubes for two terms.

Example: $a^3 - b^3$ is difference of cubes.

There is a formula too which you can use in case if you need it which goes like this:

$a^3 + b^3 = (a+b)(a^2 - ab + b^2)\\ \\ a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

Checking all options whether they result in a sum or difference of cubes

Option A: $(x-4)(x^2 + 4x - 16)$

$(x-4)(x^2 + 4x - 16) = x^3 + 64 -4x^2 + 4x^2 -16x - 16x = x^3 +4^3 -32x$

Thus, not a sum or difference of cubes

Option B: $(x-1)(x^2 - x+ 1)$

$(x-1)(x^2 - x+ 1) = x^3 - 1 + x + x -x^2 - x^2 = x^3 - 1 + 2x - 2x^2$

Thus, not a sum or difference of cubes

Option C: $(x + 1) (x - 1)$

$(x + 1) (x - 1) = x^2 - 1 + x - x = x^2 - 1$

It is difference of squares but not of cubes.

Option D: $(x+4)(x^2 -4x + 16)$

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

Thus, it is sum of cubes.

Thus, only Option D: $(x+4)(x^2 -4x + 16)$ is expressible as sum or product of cubes (here it is expressible as sum of cubes $x^3$ and $4^3$.

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