Polynomial Identities: Formulas & Proofs

What Are Polynomial Identities?

Polynomial identities are equations between two polynomial expressions that are true for every value you substitute for the variables. That "for every value" part is the whole point. An ordinary equation such as $2x + 1 = 7$ is true only for one value ($x = 3$); an identity such as $(a+b)^2 = a^2 + 2ab + b^2$ is true no matter what $a$ and $b$ are.

Both sides of an identity are the same polynomial, written two different ways. One side is usually compact and the other is expanded, which is exactly why identities are useful: they let you switch between a folded form and an unfolded form whenever one is easier to work with.

A polynomial identity is not something you solve. It is something you use, in either direction, to rewrite an expression you already have.

What Is The Difference Between An Identity And An Equation?

An identity holds for all values of the variables, while an equation holds only for the specific values that make it true. Mixing these up is the first place students slip, so it is worth pinning down.

Table: How an identity differs from an ordinary equation.

Here is a quick test. Pick random numbers and substitute. If the two sides agree for a few different choices, you probably have an identity; if they only agree for one special value, it is an equation. That substitution test is a check, not a proof, but it catches errors fast.

What Are The Standard Polynomial Identities?

Nine identities cover almost everything you meet before university. Keep this table close while the patterns settle into memory.

Table: The standard polynomial identities every student should know.

You can find each of these written out with worked cases in the Bhanzu references on algebraic identities and algebraic formulas. The rest of this article shows where each one comes from and how to use it.

How Do You Expand A Binomial Square?

To square a binomial, square the first term, square the last term, and add twice their product in the middle. That middle piece is where marks are won and lost.

$$(a+b)^2 = a^2 + 2ab + b^2$$

The $2ab$ is the middle term, and it is there because squaring $(a+b)$ means multiplying $(a+b)(a+b)$, which produces two identical cross-products, $ab$ and $ba$. The result $a^2 + 2ab + b^2$ has a name of its own, the perfect square trinomial, covered further in perfect square trinomial.

Example 1: Expand $(2x + 3)^2$.

Set $a = 2x$ and $b = 3$, then apply the identity term by term:

$$(2x)^2 = 4x^2$$

$$2 \cdot (2x) \cdot 3 = 12x$$

$$3^2 = 9$$

Final answer: $(2x + 3)^2 = 4x^2 + 12x + 9$.

The difference version flips one sign. $(a-b)^2 = a^2 - 2ab + b^2$, so only the middle term turns negative; the two squared terms stay positive. A full walk-through of the $(a+b)^2$ pattern lives at a plus b whole square.

What Is The Difference Of Squares Identity?

The difference of squares says that $a^2 - b^2$ factors into $(a+b)(a-b)$. It is the identity students reach for most, because it turns a subtraction of two squares into a clean product.

$$a^2 - b^2 = (a+b)(a-b)$$

Read left to right, it factors. Read right to left, it multiplies two binomials whose middle terms cancel. Both directions are the same truth.

Example 2: Compute $103 \times 97$ without a calculator.

Notice that $103 = 100 + 3$ and $97 = 100 - 3$, so the product is $(100+3)(100-3)$, which fits $a^2 - b^2$ with $a = 100$ and $b = 3$:

$$103 \times 97 = 100^2 - 3^2 = 10000 - 9 = 9991$$

Final answer: $103 \times 97 = 9991$, worked out in one line. The same identity factors expressions: $49x^2 - 25 = (7x + 5)(7x - 5)$, a standard first step when factoring algebraic expressions.

How Do You Expand And Factor With Cubes?

The cube identities extend the same idea one power higher, and they run in both directions, expanding and factoring. There are four to know, and their signs follow a strict pattern.

• Cube of a sum: $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

• Cube of a difference: $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$, where the signs alternate.

• Sum of cubes: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$.

• Difference of cubes: $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$.

Example 3: Expand $(x + 2)^3$.

With $a = x$ and $b = 2$, take each term of the cube-of-a-sum pattern in turn:

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

$$= x^3 + 6x^2 + 12x + 8$$

Final answer: $(x + 2)^3 = x^3 + 6x^2 + 12x + 8$.

Example 4: Factor $8x^3 + 27$.

Rewrite each part as a cube: $8x^3 = (2x)^3$ and $27 = 3^3$, so this is a sum of cubes with $a = 2x$, $b = 3$:

$$8x^3 + 27 = (2x + 3)\big((2x)^2 - (2x)(3) + 3^2\big)$$

$$= (2x + 3)(4x^2 - 6x + 9)$$

Final answer: $8x^3 + 27 = (2x + 3)(4x^2 - 6x + 9)$. The matching patterns are set out at a cube plus b cube formula and a cube minus b cube.

How Do You Handle Three Terms And Two Binomials?

Two more identities cover the cases that trip students the moment a third term or a second variable appears. The first squares a trinomial; the second multiplies two binomials that share a variable.

$$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$$

$$(x+a)(x+b) = x^2 + (a+b)x + ab$$

The trinomial square keeps every individual square and then adds twice each pair, so three terms give three cross-products, not one. The product $(x+a)(x+b)$ is the engine behind factoring quadratics: the middle coefficient is the sum $a+b$ and the constant is the product $ab$.

Example 5: Expand $(x + 5)(x + 2)$.

Here $a = 5$ and $b = 2$, so the sum is $7$ and the product is $10$:

$$(x + 5)(x + 2) = x^2 + (5 + 2)x + (5)(2) = x^2 + 7x + 10$$

Final answer: $(x + 5)(x + 2) = x^2 + 7x + 10$. Reading it backwards is how you factor $x^2 + 7x + 10$ into $(x+5)(x+2)$.

How Do You Prove A Polynomial Identity?

You prove a polynomial identity by expanding one side with the distributive law until it becomes the other side, symbol for symbol. Checking a few numbers can support your belief, but it cannot prove the identity on its own.

Take the wrong path first, because most students take it. Suppose someone claims $(a+b)^2 = a^2 + b^2$. Test $a = 1$, $b = 1$: the left side is $(1+1)^2 = 4$, the right side is $1 + 1 = 2$. The two disagree, so that claim is false, and the substitution test exposed it instantly.

Now prove the true version by expansion:

$$(a+b)^2 = (a+b)(a+b)$$

$$= a \cdot a + a \cdot b + b \cdot a + b \cdot b$$

$$= a^2 + 2ab + b^2$$

Every step is the distributive law, so the equality holds for all $a$ and $b$. That is a proof. Substituting $a = 3$, $b = 4$ and seeing $49 = 49$ is reassuring, but, actually, one lucky number would never have settled it; only the algebra does. There is also a picture-proof, the area square in the diagram above, and either one is valid.

Why Do Polynomial Identities Matter?

Polynomial identities are worth memorising because they replace slow multiplication with instant pattern-matching, and they are the reverse gear for factoring. A short list captures why they earn their place.

• Speed. Expanding $(2x+3)^2$ by the identity takes one line; multiplying it out longhand takes several and invites slips.

• Factoring. Every identity runs backwards. Difference of squares, sum and difference of cubes, and $(x+a)(x+b)$ are the first tools you try when factoring, which is the backbone of solving polynomial equations.

• A guaranteed truth. An identity holds for all values, so once proven you never re-check it. That reliability is what lets algebra build taller structures on top.

• Seeds of bigger ideas. $(a+b)^2$ and $(a+b)^3$ are the first two rows of the binomial theorem, and the square identity is the exact move inside completing the square.

Strip the shortcuts away and algebra still works, but every expansion becomes a grind and every factoring problem becomes guesswork. The identities are what make the rest of the subject fast enough to be useful.

Who Discovered Polynomial Identities?

These patterns are old, older than the symbols we write them in. For centuries mathematicians drew them as areas and volumes before anyone wrote $a$ and $b$, which is why the square-of-a-sum still has a picture attached to it.

Two more figures shaped these identities across different parts of the world:

• Euclid (around 300 BCE, Alexandria) collected geometric versions of the square and difference identities in Book II of his Elements, proving each as a statement about areas of rectangles. See the MacTutor archive on al-Khwarizmi and Euclid's work in Book II of the Elements.

• Brahmagupta (598–668 CE, India) worked with an identity for the product of two sums of squares, an early sign that identities were tools for number theory, not just geometry.

Where Are Polynomial Identities Used In The Real World?

The same handful of patterns quietly powers work far beyond the algebra classroom.

• Fast computation: mental-math methods for squaring and multiplying awkward numbers are difference-of-squares and square-of-a-sum in disguise, the $103 \times 97$ trick being the classic case.

• Computer science: the Karatsuba algorithm multiplies very large numbers faster by rearranging $(a+b)(c+d)$ so it needs three multiplications instead of four, which matters for cryptography and big-number arithmetic.

• Statistics: the spread of data uses the expansion of $(x - \bar{x})^2$, a square-of-a-difference applied across every data point.

• Physics and engineering: small-change approximations expand $(a + b)^2$ and drop the tiny $b^2$ term, a routine step when linearising a formula.

• Number theory: the identity $(a^2 - b^2)^2 + (2ab)^2 = (a^2 + b^2)^2$ generates Pythagorean triples, whole-number right triangles, straight from two starting numbers.

One short toolkit reaches from a child's mental arithmetic to cryptography and data science. That reach is a good reason to make these patterns automatic early.

What Are The Most Common Polynomial Identity Mistakes?

These four errors account for most lost marks on identities, verified against Vedantu's error notes, a Singapore secondary-school tips page, and two dedicated "common mistakes in algebra" guides.

Dropping the middle term.

Where it slips in:

A student writes $(a+b)^2 = a^2 + b^2$, treating the square like it distributes over addition.

Don't do this:

Do not forget the cross-product. Squaring a sum always produces the $2ab$ middle term.

The correct way:

Write $(a+b)^2 = a^2 + 2ab + b^2$ every time. The rusher who skips the middle term is the most common case in a first session, and the area picture is the fastest cure.

Sign slips in the cube and difference identities.

Where it slips in:

While expanding $(a-b)^3$ or $(a-b)^2$, a student loses track of which terms are negative.

Don't do this:

Do not assume every term is positive. In $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ the signs alternate.

The correct way:

Track the sign of $b$ through each term. Odd powers of a negative stay negative; even powers turn positive.

Treating an identity like an equation.

Where it slips in:

A student tries to "solve" $(a+b)^2 = a^2 + 2ab + b^2$ for a value, as if it had a solution.

Don't do this:

Do not look for the value that makes it true. It is already true for all values.

The correct way:

Use it to rewrite. An identity is a tool for expanding or factoring, not a puzzle with an answer.

Trying to factor a sum of two squares.

Where it slips in:

A student writes $a^2 + b^2 = (a+b)(a-b)$, borrowing the difference-of-squares pattern for a sum.

Don't do this:

Do not apply difference of squares to $a^2 + b^2$. Multiply $(a+b)(a-b)$ back and you get $a^2 - b^2$, not a sum.

The correct way:

Leave $a^2 + b^2$ as it is. Over the real numbers it does not factor, and recognising that is itself the correct answer.

Practice Problems On Polynomial Identities

Work each one with the matching identity, then check against the answer beside it.

1. Expand $(3x + 4)^2$.
   (Answer: $9x^2 + 24x + 16$.)

2. Expand $(2a - 5)^2$.
   (Answer: $4a^2 - 20a + 25$.)

3. Factor $81x^2 - 16$.
   (Answer: $(9x + 4)(9x - 4)$.)

4. Use difference of squares to compute $98 \times 102$.
   (Answer: $(100-2)(100+2) = 10000 - 4 = 9996$.)

5. Expand $(x - 2)^3$.
   (Answer: $x^3 - 6x^2 + 12x - 8$.)

6. Factor $27x^3 + 1$.
   (Answer: $(3x + 1)(9x^2 - 3x + 1)$.)

Where Should You Go Next After Polynomial Identities?

These patterns open straight onto the rest of algebra, and a few doors lead somewhere useful right away.

1. Factorization of algebraic expressions. Run every identity in reverse and factoring stops being guesswork.

2. The binomial theorem. See how $(a+b)^2$ and $(a+b)^3$ become a single rule for any power.

3. Polynomials. Step back to the objects these identities act on, and how they add, multiply, and divide.

If your child is building these foundations, a live Bhanzu trainer teaches identities from the picture first (the area square that makes the middle term obvious) in the Bhanzu algebra program.