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Polynomial
Identities

Learn to prove polynomial identities and apply them to describe powerful numerical relationships, including generating Pythagorean triples.

Algebra 2

What Is a Polynomial Identity?

A polynomial identity is an equation that is true for every value of the variable(s). Unlike an equation you solve for specific values, an identity holds universally.

Identity
True for all values: \((a+b)^2 = a^2 + 2ab + b^2\)
Equation
True for specific values: \(x^2 - 4 = 0\) only when \(x = \pm 2\)

Why Do Identities Matter?

Polynomial identities let us simplify complex expressions, factor difficult polynomials, and even discover hidden number theory relationships.

Simplification
Rewrite expressions into forms that are easier to compute or analyze.
Factoring
Factor sums/differences of cubes and higher powers on sight.
Number Theory
Generate infinite families of Pythagorean triples from a single formula.

Essential Identities to Know

1
Difference of Squares: \(a^2 - b^2 = (a+b)(a-b)\)
2
Perfect Square Trinomials: \((a \pm b)^2 = a^2 \pm 2ab + b^2\)
3
Sum of Cubes: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
4
Difference of Cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)
5
Pythagorean Triple Identity: \((x^2+y^2)^2 = (x^2-y^2)^2 + (2xy)^2\)

The Pythagorean Triple Identity

This remarkable identity connects algebra and geometry. For any positive integers \(x\) and \(y\) with \(x > y\), the three numbers below always form a Pythagorean triple:

$$a = x^2 - y^2, \quad b = 2xy, \quad c = x^2 + y^2$$
\(a = x^2 - y^2\)
\(b = 2xy\)
\(c = x^2 + y^2\)

How to Prove an Identity

To prove a polynomial identity, pick one side and use algebraic manipulation to transform it into the other side. You may also expand both sides independently and show they simplify to the same expression.

1
Choose a side. Start with the more complex side: it is usually easier to simplify than to build up.
2
Expand and simplify. Distribute, FOIL, or factor. Combine like terms at every step.
3
Arrive at the other side. When your result matches the other side exactly, the proof is complete.
4
State the conclusion. Write "Therefore, the identity holds for all values of the variable(s)."

Identity vs. Conditional Equation

Identity Conditional Equation
True for All values of the variable Specific values only
Goal Prove both sides are equivalent Solve for the unknown
Example \((a+b)^2 = a^2+2ab+b^2\) \(x^2 + 3x = 10\)
Technique Expand, factor, simplify Isolate the variable
Result \(LHS = RHS\) (always) \(x = 2\) or \(x = -5\)

Step-by-Step Application

Follow this checklist whenever you encounter a polynomial identity problem on an exam or assignment:

1
Identify the identity type: Is it a difference of squares, sum/difference of cubes, or the Pythagorean triple formula?
2
Write both sides: Clearly label LHS (left-hand side) and RHS (right-hand side).
3
Expand the complex side: Use distribution, FOIL, or exponent rules. Show every step.
4
Combine like terms: Simplify until you reach the other side exactly.
5
Conclude: State the identity is proven. If applying it, substitute values.

Example 1: Difference of Squares

Prove that \(a^2 - b^2 = (a+b)(a-b)\).

1
Start with the RHS: \((a+b)(a-b)\)
2
Apply FOIL: \(a \cdot a - a \cdot b + b \cdot a - b \cdot b\)
3
Simplify: \(a^2 - ab + ab - b^2 = a^2 - b^2\)
4
Conclusion: \(RHS = LHS\). The identity is proven. ✓

Example 2: Sum of Cubes

Prove that \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).

1
Start with the RHS: Expand \((a+b)(a^2 - ab + b^2)\)
2
Distribute \(a\): \(a^3 - a^2b + ab^2\)
3
Distribute \(b\): \(a^2b - ab^2 + b^3\)
4
Combine: \(a^3 \cancel{- a^2b} + \cancel{ab^2} + \cancel{a^2b} \cancel{- ab^2} + b^3 = a^3 + b^3\) ✓

Example 3: Pythagorean Triple Proof

Prove: \((x^2+y^2)^2 = (x^2-y^2)^2 + (2xy)^2\)

1
Expand RHS first term: \((x^2-y^2)^2 = x^4 - 2x^2y^2 + y^4\)
2
Expand RHS second term: \((2xy)^2 = 4x^2y^2\)
3
Add them: \(x^4 - 2x^2y^2 + y^4 + 4x^2y^2 = x^4 + 2x^2y^2 + y^4\)
4
Recognize LHS: \((x^2+y^2)^2 = x^4 + 2x^2y^2 + y^4\). Both sides match! ✓

Generating Pythagorean Triples

Now let's apply the identity. Choose any positive integers \(x > y\), then compute:

$$a = x^2 - y^2, \quad b = 2xy, \quad c = x^2 + y^2$$
\(x\) \(y\) \(a = x^2 - y^2\) \(b = 2xy\) \(c = x^2 + y^2\)
2 1 3 4 5
3 2 5 12 13
4 1 15 8 17
4 3 7 24 25

Visualizing a Proof Flow

Choose a Side
(pick the complex one)
Expand & Simplify
(FOIL, distribute)
Combine Terms
(cancel like terms)
Does it match the other side?
(compare term by term)
Identity Proven ✓
Not yet? Keep simplifying
(or try the other side)

Example 4: Numerical Application

Use the difference of squares identity to compute \(47^2 - 53^2\) without a calculator.

1
Recognize the form: \(47^2 - 53^2 = a^2 - b^2\) with \(a = 47\), \(b = 53\)
2
Apply the identity: \((a+b)(a-b) = (47+53)(47-53)\)
3
Compute: \((100)(-6) = -600\)

Common Mistakes

✕ Confusing Identity with Equation
Trying to "solve for \(x\)" instead of proving both sides are equivalent for all values.
✕ Sign Errors in Expansion
Forgetting that \((a-b)^2 = a^2 - 2ab + b^2\), not \(a^2 - b^2\). The middle term is critical.
✕ Wrong Cube Factorization
Mixing up \(a^3 + b^3\) and \(a^3 - b^3\). Remember: the binomial factor matches the sign, the trinomial factor has the opposite middle term.

Identity Quick Reference

Identity Name Formula Key Use
Difference of Squares \(a^2 - b^2 = (a+b)(a-b)\) Fast mental arithmetic
Perfect Square \((a \pm b)^2 = a^2 \pm 2ab + b^2\) Expanding / completing the square
Sum of Cubes \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\) Factoring cubic expressions
Diff. of Cubes \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\) Factoring cubic expressions
Pythagorean Triple \((x^2+y^2)^2 = (x^2-y^2)^2 + (2xy)^2\) Generating integer right triangles

Question 1 of 5

Which of the following is a valid polynomial identity?

Question 2 of 5

Using the Pythagorean triple identity with \(x = 3\) and \(y = 1\), what triple is generated?

Question 3 of 5

When proving the identity \((a+b)^2 = a^2 + 2ab + b^2\), what is the correct first step?

Question 4 of 5

Which factorization correctly represents the sum of cubes identity?

Question 5 of 5

Using the difference of squares identity, what is the value of \(101^2 - 99^2\)?