Example

Does [latex](a+b)(a^{2}–ab+b^{2})=a^{3}+b^{3}[/latex]?

Answer: Apply the distributive property.

[latex]\left(a\right)\left(a^{2}–ab+b^{2}\right)+\left(b\right)\left(a^{2}–ab+b^{2}\right)[/latex]

Multiply by a.

[latex]\left(a^{3}–a^{2}b+ab^{2}\right)+\left(b\right)\left(a^{2}-ab+b^{2}\right)[/latex]

Multiply by b.

[latex]\left(a^{3}–a^{2}b+ab^{2}\right)+\left(a^{2}b–ab^{2}+b^{3}\right)[/latex]

Rearrange terms in order to combine the like terms.

[latex]a^{3}-a^{2}b+a^{2}b+ab^{2}-ab^{2}+b^{3}[/latex]

Simplify.

[latex]a^{3}+b^{3}[/latex]

Example

Factor [latex]x^{3}+8y^{3}[/latex].

Answer: Identify that this binomial fits the sum of cubes pattern [latex]a^{3}+b^{3}[/latex]. [latex]a=x[/latex], and [latex]b=2y[/latex] (since [latex]2y\cdot2y\cdot2y=8y^{3}[/latex]).

[latex]x^{3}+8y^{3}[/latex]

Factor the binomial as [latex]\left(a+b\right)\left(a^{2}–ab+b^{2}\right)[/latex], substituting [latex]a=x[/latex] and [latex]b=2y[/latex] into the expression.

[latex]\left(x+2y\right)\left(x^{2}-x\left(2y\right)+\left(2y\right)^{2}\right)[/latex]

Square [latex](2y)^{2}=4y^{2}[/latex].

[latex]\left(x+2y\right)\left(x^{2}-x\left(2y\right)+4y^{2}\right)[/latex]

Multiply [latex]−x\left(2y\right)=−2xy[/latex] (writing the coefficient first).

The factored form is [latex]\left(x+2y\right)\left(x^{2}-2xy+4y^{2}\right)[/latex].

Example

Factor [latex]16m^{3}+54n^{3}[/latex].

Answer: Factor out the common factor [latex]2[/latex].

[latex]16m^{3}+54n^{3}[/latex]

[latex]2\left(8m^{3}+27n^{3}\right)[/latex]

[latex]8m^{3}[/latex] and [latex]27n^{3}[/latex] are cubes, so you can factor [latex]8m^{3}+27n^{3}[/latex] as the sum of two cubes: [latex]a=2m[/latex] and [latex]b=3n[/latex]. Factor the binomial [latex]8m^{3}+27n^{3}[/latex] substituting [latex]a=2m[/latex] and [latex]b=3n[/latex] into the expression [latex]\left(a+b\right)\left(a^{2}-ab+b^{2}\right)[/latex].

[latex]2\left(2m+3n\right)\left[\left(2m\right)^{2}-\left(2m\right)\left(3n\right)+\left(3n\right)^{2}\right][/latex]

Square: [latex](2m)^{2}=4m^{2}[/latex] and [latex](3n)^{2}=9n^{2}[/latex].

[latex]2\left(2m+3n\right)\left[4m^{2}-\left(2m\right)\left(3n\right)+9n^{2}\right][/latex]

Multiply [latex]-\left(2m\right)\left(3n\right)=-6mn[/latex].

The factored form is [latex]2\left(2m+3n\right)\left(4m^{2}-6mn+9n^{2}\right)[/latex].

Example

Factor [latex]8x^{3}–1,000[/latex].

Answer: Factor out [latex]8[/latex].

[latex]8(x^{3}–125)[/latex]

Identify that the binomial fits the pattern [latex]a^{3}-b^{3}:a=x[/latex], and [latex]b=5[/latex] (since [latex]5^{3}=125[/latex]). Factor [latex]x^{3}–125[/latex] as [latex]\left(a–b\right)\left(a^{2}+ab+b^{2}\right)[/latex], substituting [latex]a=x[/latex] and [latex]b=5[/latex] into the expression.

[latex]8\left(x-5\right)\left[x^{2}+\left(x\right)\left(5\right)+5^{2}\right][/latex]

Square the first and last terms, and rewrite [latex]\left(x\right)\left(5\right)[/latex] as [latex]5x[/latex].

[latex]8\left(x–5\right)\left(x^{2}+5x+25\right)[/latex]

Example

Factor [latex]r^{9}-8s^{6}[/latex].

Answer: Identify this binomial as the difference of two cubes. As shown above, it is. Rewrite [latex]r^{9}[/latex] as [latex]\left(r^{3}\right)^{3}[/latex] and rewrite [latex]8s^{6}[/latex] as [latex]\left(2s^{2}\right)^{3}[/latex].

[latex]\left(r^{3}\right)^{3}-\left(2s^{2}\right)^{3}[/latex]

Now the binomial is written in terms of cubed quantities. Thinking of [latex]a^{3}-b^{3}[/latex], [latex]a=r^{3}[/latex] and [latex]b=2s^{2}[/latex]. Factor the binomial as [latex]\left(a-b\right)\left(a^{2}+ab+b^{2}\right)[/latex], substituting [latex]a=r^{3}[/latex] and [latex]b=2s^{2}[/latex] into the expression.

[latex]\left(r^{3}-2s^{2}\right)\left[\left(r^{3}\right)^{2}+\left(r^{3}\right)\left(2s^{2}\right)+\left(2s^{2}\right)^{2}\right][/latex]

Multiply and square the terms.

[latex]\left(r^{3}-2s^{2}\right)\left(r^{6}+2r^{3}s^{2}+4s^{4}\right)[/latex]