Factoring by Substitution

The temporary variable method-often called substitution-is a powerful technique for simplifying the structure of a polynomial so that familiar factoring tools can be applied.

It is particularly effective when a quadratic pattern is “hidden” inside a higher-degree expression.

How does it work?

Choose part of the expression to replace with a temporary variable, for instance: $$ t = x^2, \quad t = x^3, \quad \text{or} \quad t = x^2 + 1 $$
Rewrite the polynomial in terms of the new variable $t$.
Apply a standard factoring method (discriminant, sum and product, or special product identities).
Substitute the original expression back in place of $t$.

The aim is to reduce a higher-degree polynomial to the familiar quadratic form, where the usual factorization techniques can be applied directly.

Note. This method only works if all terms fit consistently into the same substitution. If the degrees don’t align or the terms cannot be expressed uniformly, substitution won’t simplify the problem. It is especially effective for symmetric polynomials or those involving even powers such as $x^4, x^2,$ and constants.

A Worked Example

Example 1

Consider the polynomial:

$$ x^4 + 5x^2 + 6 $$

Introduce the substitution $ t = x^2 $:

$$ x^4 + 5x^2 + 6 = t^2 + 5t + 6 $$

Now we have a quadratic in $t$, which can be factored easily using the sum-product method:

$$ t^2 + 5t + 6 = (t + 2)(t + 3) $$

Note. For a quadratic of the form $t^2 + bt + c$, if two numbers $p$ and $q$ exist such that $p+q = b$ and $pq = c$, then the polynomial factors as $ (t+p)(t+q) $. In this case, $p = 2$ and $q = 3$ satisfy the conditions for $t^2 + 5t + 6$.

Replacing $t$ with the original variable gives:

$$ (t + 2)(t + 3) = (x^2 + 2)(x^2 + 3) $$

So the factorization is:

$$ (x^2 + 2)(x^2 + 3) $$

Example 2

Now take a polynomial quadratic in $x^3$:

$$ x^6 - 7x^3 + 10 $$

Substitute $ t = x^3 $:

$$ t^2 - 7t + 10 $$

This is a straightforward quadratic in $t$, which factors as:

$$ t^2 - 7t + 10 = (t - 2)(t - 5) $$

Finally, restore the original variable:

$$ (t - 2)(t - 5) = (x^3 - 2)(x^3 - 5) $$

Therefore, the factorization is:

$$ (x^3 - 2)(x^3 - 5) $$

The same approach can be extended to many other cases.