Factoring the Sum and Difference of Cubes
Now we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.
Sum and Difference of Cubes
The sum and difference of cubes are algebraic formulas used to factor and solve polynomial equations involving cubic terms.
- Sum of Cubes: [latex]{a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)[/latex]
- Difference of Cubes: [latex]{a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)[/latex]
- Confirm that the first and last term are cubes, [latex]{a}^{3}+{b}^{3}[/latex] or [latex]{a}^{3}-{b}^{3}[/latex].
- For a sum of cubes, write the factored form as [latex]\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)[/latex]. For a difference of cubes, write the factored form as [latex]\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)[/latex].
To factor the cubic expression [latex]{x}^{3}+512[/latex], we recognize that [latex]512[/latex] is a perfect cube, specifically [latex]8^3[/latex]. This allows us to apply the sum of cubes formula.
[latex]\begin{align*} \text{Original expression:} & \quad x^3 + 512 \\ \text{Identify cubes:} & \quad x^3 \text{ and } 512 = 8^3 \\ \text{Apply sum of cubes formula:} & \quad x^3 + 8^3 = (x + 8)(x^2 - 8x + 64) \\ \text{Factored form:} & \quad (x + 8)(x^2 - 8x + 64) \end{align*}[/latex]
Factoring Expressions with Fractional or Negative Exponents
Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, [latex]2{x}^{\frac{1}{4}}+5{x}^{\frac{3}{4}}[/latex] can be factored by pulling out [latex]{x}^{\frac{1}{4}}[/latex] and being rewritten as [latex]{x}^{\frac{1}{4}}\left(2+5{x}^{\frac{1}{2}}\right)[/latex].