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Factor a cubic expression by substituting x plus one

Replace a repeated expression with one variable

When a polynomial repeatedly contains the same expression, such as $x + 1$ , temporarily call that expression $t$ . Factor in $t$ , then substitute back.

Problem

Factor $(x + 1)^{3} - 3 (x + 1) + 2$ .

Answer

x^{2} (x + 3)

The factor x occurs twice.

Step-by-step solution

1. Make the substitution

Let $t = x + 1$ . The expression becomes $t^{3} - 3 t + 2$ .

2. Find a factor

At $t = 1$ , the polynomial equals $1 - 3 + 2 = 0$ . Therefore $t - 1$ is a factor. Division gives

t^{3} - 3 t + 2 = (t - 1) (t^{2} + t - 2) .

3. Finish factoring and substitute back

The quadratic factors as $(t - 1) (t + 2)$ . Thus the full factorization is $(t - 1)^{2} (t + 2)$ . Since $t = x + 1$ , the result is $x^{2} (x + 3)$ .

4. Check by expansion

Direct expansion of the original gives $x^{3} + 3 x^{2}$ , which is also $x^{2} (x + 3)$ .

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