Question

Explain how you can prove the difference of two cubes identity. a^3 – b^3 = (a – b)(a^2 + ab + b^2)

Answer 1

Answer:

Use the distributive property to multiply the factors on the right side of the equation.

Simplify the product by combining like terms.

Show that the right side of the equation can be written exactly the same as the left side.

Show that the right side of the equation simplifies to a cubed minus b cubed.

Step-by-step explanation:

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Answer 2

A polynomial in the form $\mathbf{a^3 - b^3}$ is called a difference of cubes.

The equation is given as:

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

Apply distributive property

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

Expand

$\mathbf{a^3 - b^3 = a^3 + a^2b + ab^2- a^2b - ab^2 - b^3}$

Rearrange: Collect like terms

$\mathbf{a^3 - b^3 = a^3 + a^2b - a^2b+ ab^2 - ab^2 - b^3}$

Evaluate like terms

$\mathbf{a^3 - b^3 = a^3 - b^3}$

Hence, the difference of cubes identity has been proved.

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