Four patterns, recognised rather than derived. Each is one of the special products of “Special Products” read backwards, and each is worth knowing on sight because the general methods are slower and, for the cubes, will not find the factorisation at all.
Core ideas
These are identities, so they hold in both directions. Reading them right to left is multiplying; reading them left to right is factoring.
9x² - 25 is a difference of squares because 9x² is (3x)² and 25 is 5². Over the rational coefficients used here, 9x² - 24 does not match because 24 is not a rational square. Over a larger coefficient system the answer can change, so the qualifier is part of the claim.
The identity needs a² - b²; changing the minus to a plus destroys the middle-term cancellation. In particular x² + b² with nonzero real b has no real linear factor because it is always positive. Higher-degree sums of square expressions require separate analysis; the plus sign alone is not a universal irreducibility test.
x⁴ - 16 is (x² - 4)(x² + 4), and x² - 4 is itself a difference of squares. A pattern that applies once may apply again to one of the pieces, and an answer that stops early is incomplete.
The material
How to recognise each, and the two that are most often faked.
“Special Products” multiplied these out. Every identity here is one of those products with the two sides exchanged, which is why they need no proof beyond expanding the right-hand side.
Expanding the right side gives a² + ab - ab - b² = a² - b²; the middle terms cancel. Conversely, any expression that is one perfect square minus another factors this way.
9x² - 25 = (3x)² - 5² = (3x - 5)(3x + 5). And x⁴ - 16 = (x²)² - 4² = (x² - 4)(x² + 4) = (x - 2)(x + 2)(x² + 4), where the last factor is a sum of squares and stops there.
There is no difference-of-squares pattern for a sum. a² + b² is not (a - b)(a + b). In the quadratic example x² + 4, every real x gives x² + 4 > 0, so there is no real root and therefore no real linear factor. Writing x² + 4 = (x + 2)(x + 2) is the single most common invented law in this lesson; expanding it gives x² + 4x + 4.
A trinomial is a perfect square exactly when its first and last terms are perfect squares and its middle term is twice the product of their roots, up to sign.
For the first, x² = (x)², 36 = 6², and 2 · x · 6 = 12x matches: it is (x + 6)². For the second, 25x² = (5x)², 4 = 2², and 2 · 5x · 2 = 20x matches the middle term with the sign negative: it is (5x - 2)².
The middle-term test is what makes this a theorem rather than a guess. x² + 13x + 36 has square outer terms too, but 2 · 1 · 6 = 12, not 13, so it is not a perfect square. It happens to factor as (x + 4)(x + 9), by the method of “Factoring Simple Trinomials”.
Unlike squares, a SUM of cubes does factor. In both identities the linear factor carries the sign of the original and the quadratic factor carries the opposite sign on its middle term. Retest that factor like every other one; the cube identity alone does not prove that no different pattern applies to it.
8x³ - 27 = (2x)³ - 3³ = (2x - 3)(4x² + 6x + 9). And 27x³ + 64 = (3x)³ + 4³ = (3x + 4)(9x² - 12x + 16). Expanding either one back is four lines and worth doing once.
The sign rule is the part that is misremembered. In a³ - b³ the binomial is (a - b) and the trinomial's middle term is +ab; in a³ + b³ the binomial is (a + b) and the middle term is -ab. The trinomial's last term is +b² in both cases, never negative.
Interactive
Four patterns tested against what you typed
Method
Count the terms first; the count tells you which patterns are even possible.
2x² - 50 is not a difference of squares as written, because 2x² is not a perfect square. As 2(x² - 25) it is, giving 2(x - 5)(x + 5).
A difference of rational perfect squares matches the conjugate identity; a sum of squares does not match that identity and needs a separate factorability argument. Both a sum and a difference of rational perfect cubes match their cube identities. Check the powers before naming a pattern.
If the outer terms are perfect squares, compute twice the product of their roots. If that equals the middle term, it is a perfect square trinomial; if not, use “Factoring Simple Trinomials” or “The AC Method”.
Apply the list to every factor you produced. Stop only when no factor matches any pattern and none factors by the AC method.
Two terms, both perfect squares, so the difference of squares applies — and then applies again.
Two things are being claimed at the end, and only one of them is a calculation. That (x - 2)(x + 2)(x² + 4) multiplies back to x⁴ - 16 is checked by expanding. That x² + 4 cannot be factored further is a separate claim, and its justification is that a sum of squares has no real linear factor because it stays positive — not that no factorisation came to mind.
For a faded special-form pass, start 16x⁴ - 81 with the supplied recognition (4x²)² - 9². Apply the pattern, retest both factors over the rationals, and expand to check. The complete result is (2x - 3)(2x + 3)(4x² + 9); stopping at (4x² - 9)(4x² + 9) misses the second difference of squares.
Given 9x² - 25, x² + 12x + 36, 8x³ - 27 and x² + 16 you should factor the first three on sight and say of the fourth that the conjugate pattern does not apply; because it stays positive, it also has no real linear factor. Neither conclusion is a failure to spot something.
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