HSA.APR.C.4: Proving Polynomial Identities and Using Them with Numbers
In plain English: HSA.APR.C.4 is the Common Core algebra standard that asks students to prove polynomial identities, such as a² - b² = (a + b)(a - b), by rewriting one side until it matches the other, and then to use those identities to explain numerical patterns. The official example uses (x² + y²)² = (x² - y²)² + (2xy)² to generate Pythagorean triples. It is usually taught in Algebra II.
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.
Common Core State Standards for Mathematics · Domain: Arithmetic with Polynomials and Rational Expressions (APR) · Cluster: Use polynomial identities to solve problems Also written as HSA-APR.C.4 or A-APR.4 · Official standard
Students learn what makes an equation a polynomial identity: it is true for every value of its variables, not only for a few. They prove identities by expanding and regrouping one side, using the distributive, commutative and associative properties, until it matches the other side, and they disprove false claims with a single counterexample. Checking a few numbers is treated as evidence, never as proof.
Students then use identities as tools for numbers. The difference of squares turns 38 · 42 into 40² - 2², the identity (n + 1)² - n² = 2n + 1 explains why every odd number is a difference of consecutive squares, and the official example (x² + y²)² = (x² - y²)² + (2xy)² produces as many Pythagorean triples as students want.
Learning Objectives
By the end of this lesson, students will be able to:
Explain the difference between an equation that is true for some values and a polynomial identity that is true for all values
Prove a polynomial identity by transforming one side, or both sides, into the same expression using properties of operations
Disprove a claimed identity with a counterexample
Prove the identity (x² + y²)² = (x² - y²)² + (2xy)² and use it to generate Pythagorean triples
Use identities such as the difference of squares and the sum of cubes to calculate mentally and to explain numerical patterns
Prior Knowledge Required
Students should already be comfortable with:
Adding, subtracting and multiplying polynomials HSA.APR.A.1
Rewriting expressions by seeing their structure, such as x⁴ - y⁴ as (x²)² - (y²)² HSA.SSE.A.2
Write two claims on the board and give students three minutes to test each with at least four numbers, including a negative number and a fraction.
Warm-Up Prompt
"Claim 1: n² + n + 41 is a prime number for every whole number n. Claim 2: n² + 2n + 1 is a perfect square for every whole number n. Test each claim. Which one can you be sure about, and why?"
Claim 1 gives primes for n = 0, 1, 2, 3 and far beyond (41, 43, 47, 53, ...), so many students will believe it. Then show n = 40: 40² + 40 + 41 = 1681 = 41², which is not prime. Claim 2 cannot be settled by testing either, but algebra settles it at once: n² + 2n + 1 = (n + 1)² for every n. Use the contrast to state the lesson's rule: examples can disprove a claim, only an algebraic argument proves one.
Direct Instruction20-25 minutes
What an identity is. An equation such as x² = 9 is true for two values of x. An identity, such as x² - 9 = (x - 3)(x + 3), is true for every value. To prove an identity, follow these steps:
Pick a starting side: usually the more complicated one, because expanding is easier than factoring.
Rewrite it one justified step at a time: distribute, combine like terms, or use a known identity such as (a + b)² = a² + 2ab + b².
Stop when it matches the other side exactly, or rewrite both sides until they reach the same expression.
Never move terms across the equals sign: that assumes the identity is already true.
To disprove a claim, give one set of values where the two sides differ.
Equation: Every odd number 2n + 1 is a difference of consecutive squares, for example 15 = 8² - 7²
After the third example, show Diagram 1: the legs x² - y² and 2xy and the hypotenuse x² + y² for x = 3, y = 2, drawn to scale. After the second example, use Diagram 2 to show why a² - b² = (a + b)(a - b): cutting the L-shaped region and moving one piece turns it into an (a + b)-by-(a - b) rectangle. Stress that the picture explains the identity for positive lengths, while the algebra proves it for all numbers.
Guided Practice15 minutes
Work through two proofs with the class, asking students to justify each line. (1) Prove x³ - y³ = (x - y)(x² + xy + y²) by distributing the right side: x³ + x²y + xy² - x²y - xy² - y³ = x³ - y³. Then use it: 10³ - 7³ = 3(100 + 70 + 49) = 3 · 219 = 657, which matches 1000 - 343. (2) Prove (a² + b²)(c² + d²) = (ac - bd)² + (ad + bc)² by expanding both sides to a²c² + a²d² + b²c² + b²d². Then use it with 5 = 1² + 2² and 13 = 2² + 3²: 65 = (2 - 6)² + (3 + 4)² = 16 + 49. Ask: what does the identity say about any product of two sums of two squares? Watch for these errors: moving terms across the equals sign during a proof, (a - b)² written as a² - b², and dropping the middle term 2ab.
Independent Practice15 minutes
Students work alone on four tasks, then compare with a partner. (1) Prove (a + b)² = a² + 2ab + b² by distributing (a + b)(a + b), then use it to compute 103² = 10,000 + 600 + 9 = 10,609. (2) Use the difference of squares to compute 97 · 103 = 100² - 3² = 9991. (3) Use the official identity with x = 4, y = 1 to produce the triple 15, 8, 17 and check that 225 + 64 = 289. (4) Decide whether (x - 2)² = x² - 4 is an identity. (It is not: x = 1 gives 1 on the left and -3 on the right.)
Closure5-10 minutes
Exit ticket: (1) A classmate checks x = 0, 1 and 2 and says an identity is proved. Explain in one sentence why that is not a proof. (2) Prove (x + 3)² - (x - 3)² = 12x. (Both expansions give x² ± 6x + 9, and the difference is 12x.) (3) Use an identity to compute 59 · 61. (60² - 1 = 3599.)
Differentiation Strategies
For Struggling Students
Give a proof template with two columns: the expression on each line and the property used, such as "distributive property" or "combine like terms"
Let students test a claim with two or three numbers first, then prove it, so they see that testing and proving answer different questions
Keep a reference card with the three core identities: (a + b)², (a - b)² and a² - b²
For Advanced Students
Prove x⁴ + 4 = (x² + 2x + 2)(x² - 2x + 2) and use it to show that n⁴ + 4 is never prime for a whole number n greater than 1
Show that every Pythagorean triple from the official identity with x and y of opposite parity and no common factor is primitive, and test several cases
Find two different ways to write 1105 = 5 · 13 · 17 as a sum of two squares using the product identity from Guided Practice
Assessment Guidance
What to Look For
A complete proof starts from one side and reaches the other through steps that each keep the expression equal, with no terms moved across the equals sign. Ask students to name the property behind any step you question. When students use an identity with numbers, check that they name the identity and the values of the variables (for example, "a² - b² with a = 40, b = 2"), not only the final number. A counterexample must include both sides evaluated, so the reader can see they differ.
02
Classroom Activities
3 Activities
1
Identity or Impostor? Card Sort
20 minGroups of 3
Each group gets 8 cards, each with a claimed identity. Groups sort them into "identity" and "impostor". Every impostor needs a counterexample and every identity needs a written proof.
The 8 Cards
(x + 2)² = x² + 4 (impostor: x = 1 gives 9 and 5)
x² - 9 = (x - 3)(x + 3) (identity)
(a - b)² = (b - a)² (identity)
(x + 1)³ = x³ + 1 (impostor: x = 1 gives 8 and 2)
x³ + 8 = (x + 2)(x² - 2x + 4) (identity)
(2x)² = 2x² (impostor: x = 1 gives 4 and 2)
(x² + 1)(x + 1) = x³ + x² + x + 1 (identity)
(a + b)(a - b) + b² = a² (identity)
Procedure
Each student tests every card with one number of their choice and records the result
The group sorts the cards, then splits the proofs and counterexamples among its members
Members swap papers and check each other's work line by line
Discussion Questions
Card 4 is true for x = 0. Why does that not make it an identity?
Which impostors come from "distributing" an exponent over a sum?
Could a claim pass 100 number tests and still be false? Recall the warm-up.
2
Pythagorean Triple Factory
20 minPairs
Pairs use the official example (x² + y²)² = (x² - y²)² + (2xy)² as a machine: choose whole numbers x > y > 0, and the identity outputs a right triangle with whole-number sides.
Procedure
Prove the identity first (each partner expands one side), so the factory is guaranteed to work
Fill a table with columns x, y, x² - y², 2xy, x² + y² for these pairs: (2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 2). The outputs are 3-4-5, 8-6-10, 5-12-13, 15-8-17, 12-16-20, 7-24-25 and 21-20-29
Check each row with the Pythagorean Theorem and draw one triangle to scale on grid paper
Discussion Questions
Why must x be greater than y?
Which rows give triples that are multiples of a smaller triple? What do those (x, y) pairs have in common?
The identity gives a triple for every input pair. Does it prove anything about triangles that are not right triangles?
Challenge Variation
Ask pairs to find inputs (x, y) that give a triple with hypotenuse 65, and to explain why there are two different answers (x = 8, y = 1 and x = 7, y = 4).
3
Number Tricks, Explained
20 minPairs, then whole class
The teacher performs three "mind-reading" number tricks. Pairs try each trick with their own numbers, write it as a polynomial expression and prove the identity that makes it work.
The Tricks
Trick 1: Add 1 to your number and square it, subtract 1 from your number and square it, and subtract. You always get 4 times your number: (n + 1)² - (n - 1)² = 4n
Trick 2: Multiply two numbers that differ by 2 and add 1. You always get a perfect square: n(n + 2) + 1 = (n + 1)²
Trick 3: Multiply four consecutive whole numbers and add 1. You always get a perfect square: n(n + 1)(n + 2)(n + 3) + 1 = (n² + 3n + 1)². For example, 2 · 3 · 4 · 5 + 1 = 121 = 11²
Procedure
Pairs test each trick with three numbers, including one negative number
Pairs write the trick with a variable and prove the identity; for Trick 3, multiply n(n + 3) and (n + 1)(n + 2) first
Each pair invents a new trick based on an identity from the lesson and tries it on another pair
Modification for Distance Learning
Post the tricks on a shared slide. Students submit their test results in a shared table and upload a photo of one proof.
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Diagrams & Visual Aids
2 diagrams
Diagram 1: The Official Identity as a Triple Generator
For x = 3 and y = 2 the identity (x² + y²)² = (x² - y²)² + (2xy)² gives legs 5 and 12 and hypotenuse 13; the triangle is drawn to scale (20 px per unit). The table shows the outputs for four input pairs.
Diagram 2: Why a² - b² = (a + b)(a - b)
Remove a b-by-b square from an a-by-a square (here a = 7, b = 3, drawn to scale). Piece B, which is (a - b) by b, moves next to piece A, which is a by (a - b). Together they form an (a + b)-by-(a - b) rectangle with the same area.
04
Homework Assignment
~30 min
HSA.APR.C.4 Homework: Proving and Using Polynomial Identities
Directions: Write every proof as a chain of equal expressions, starting from one side and ending at the other, and name the property or identity you use in each step. Do not move terms across the equals sign in a proof. Use a calculator only to check a numerical answer after you have found it with an identity.
Part 1: Proving Identities (Problems 1-3)
Prove that (x + y)² + (x - y)² = 2(x² + y²). Then use the identity with x = 10 and y = 3 to find 13² + 7² without squaring 13 or 7.
Prove that a³ + b³ = (a + b)(a² - ab + b²). Use the identity to explain why 5³ + 3³ is divisible by 8, and find the other factor.
Decide whether each equation is an identity. Prove the identities and give a counterexample for the others: (a) (x - 1)(x² + x + 1) = x³ - 1 (b) (x + 5)² = x² + 25 (c) (x + y)² - 4xy = (x - y)²
Part 2: Identities and Numbers (Problems 4-6)
Use the difference of squares identity to compute (a) 64 · 56 and (b) 205 · 195 without a calculator. For each, state the values of a and b you used.
Use the identity (x² + y²)² = (x² - y²)² + (2xy)² with (a) x = 5, y = 4 and (b) x = 7, y = 2 to produce two Pythagorean triples. Verify each with the Pythagorean Theorem, and explain why the identity needs x > y.
Prove that the difference of the squares of two consecutive odd numbers is always a multiple of 8. (Hint: write the odd numbers as 2n - 1 and 2n + 1.) Check your result with 11² - 9².
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Proof Structure
Chain of equal expressions from one side to the other, each step justified
Correct algebra but steps not justified, or terms moved across the equals sign
Answer each question, then open the explanation. Your score updates as you go, and Reset quiz clears all answers so you can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which equation is a polynomial identity?
Answer: B
Expanding (x - 4)(x + 4) gives x² + 4x - 4x - 16 = x² - 16 for every x. Choice C is an equation true only for x = 4 and x = -4. Choices A and D forget the middle term: (x + 4)² = x² + 8x + 16 and (x - 4)² = x² - 8x + 16.
Question 2 of 20 · Multiple Choice
Which argument proves that (x + 5)(x - 5) = x² - 25 is an identity?
Answer: D
Transforming the left side into the right side with the distributive property works for every x, so it is a proof. Choices A and B check only some values, which can never prove a statement about all values. Choice C starts by assuming the equation is true and changes both sides, and it also skips the expansion.
Question 3 of 20 · Multiple Choice
Which values of a and b are a counterexample to the claim (a + b)² = a² + b²?
Answer: B
With a = 2, b = 3: (2 + 3)² = 25 but 2² + 3² = 13. The sides differ, so the claim is false. In choices A, C and D one of the numbers is 0, so the missing term 2ab is 0 and both sides agree: those values do not disprove the claim.
Question 4 of 20 · Multiple Choice
Use a difference of squares to compute 31 · 29.
Answer: A
31 · 29 = (30 + 1)(30 - 1) = 30² - 1² = 900 - 1 = 899. Choice B adds 1 instead of subtracting it. Choice C forgets the b² term. Choice D is 31².
Question 5 of 20 · Multiple Choice
Which is the expansion of (2x + 3)²?
Answer: C
Use (a + b)² = a² + 2ab + b² with a = 2x, b = 3: 4x² + 2(2x)(3) + 9 = 4x² + 12x + 9. Choice A drops the 2ab term. Choice B uses ab instead of 2ab. Choice D squares only x, not 2x.
Question 6 of 20 · Multiple Choice
The identity (x² + y²)² = (x² - y²)² + (2xy)² with x = 6 and y = 1 produces which Pythagorean triple?
Answer: A
x² - y² = 36 - 1 = 35, 2xy = 12 and x² + y² = 37. Check: 1225 + 144 = 1369 = 37². Choice B uses x² instead of x² - y² for the first leg, and 36² + 12² = 1440 is not 37². Choice D is the triple for x = 3, y = 2.
Question 7 of 20 · Multiple Choice
Which input pair (x, y) makes the official identity produce the triple 11, 60, 61?
Answer: D
With x = 6, y = 5: x² - y² = 11, 2xy = 60 and x² + y² = 61. Choice A gives 24, 10, 26 and choice B gives 91, 60, 109. Choice C plugs in the legs themselves, which makes x² - y² negative (-3479).
Question 8 of 20 · Multiple Choice
Which factorization of x³ - 27 is correct?
Answer: B
By the identity x³ - y³ = (x - y)(x² + xy + y²) with y = 3: (x - 3)(x² + 3x + 9). Expanding gives x³ + 3x² + 9x - 3x² - 9x - 27 = x³ - 27. Choice A has the wrong sign on the middle term; it expands to x³ - 6x² + 18x - 27. Choice C expands to x³ - 9x² + 27x - 27.
Question 9 of 20 · Multiple Choice
The identity (2n + 1)² = 4(n² + n) + 1 explains which fact about numbers?
Answer: C
Every odd number can be written 2n + 1, and its square is 4(n² + n) + 1: a multiple of 4, plus 1. For example, 7² = 49 = 4 · 12 + 1. Choice A is false: 9 is not a multiple of 4. Choice B is false: 4 = 2² is itself a multiple of 4, not one more than one. Choice D is false: 9 + 1 = 10 is not a multiple of 4.
Question 10 of 20 · Multiple Choice
Which equation is NOT an identity?
Answer: C
x = 1 gives (1 + 3)² = 16 on the left and 1 + 9 = 10 on the right, so choice C is not an identity: it drops the middle term 6x. Choices A, B and D are all true for every x, which you can show by distributing.
Question 11 of 20 · Multiple Choice
Use the sum of cubes identity a³ + b³ = (a + b)(a² - ab + b²) to decide which number must divide 12³ + 8³.
Answer: A
With a = 12, b = 8, the identity shows that 12³ + 8³ has the factor a + b = 20. Indeed 1728 + 512 = 2240 = 20 · 112. Choice B, 3, is a common guess because 12 is a multiple of 3, but 2240 is not a multiple of 3. Neither 9 nor 11 divides 2240.
Question 12 of 20 · Multiple Choice
Which is the expansion of (a - b)³?
Answer: D
Multiply (a - b)² = a² - 2ab + b² by (a - b): a³ - 2a²b + ab² - a²b + 2ab² - b³ = a³ - 3a²b + 3ab² - b³. Choice A drops the middle terms; a = 2, b = 1 gives 1 for (a - b)³ but 7 for a³ - b³. Choices B and C have sign errors.
Question 13 of 20 · Multiple Choice
Which expression is equal to n(n + 6) + 9 for every n?
Answer: A
Expand: n(n + 6) + 9 = n² + 6n + 9 = (n + 3)². So the product of two numbers that differ by 6, plus 9, is always a perfect square, for example 4 · 10 + 9 = 49 = 7². Choice C forgets the constant 9. Choice B drops the 6n term, and choice D drops the n² term.
Question 14 of 20 · Multiple Choice
Use (a² + b²)(c² + d²) = (ac - bd)² + (ad + bc)² with 10 = 1² + 3² and 13 = 2² + 3² to write 130 as a sum of two squares.
Answer: D
Here a = 1, b = 3, c = 2, d = 3: ac - bd = 2 - 9 = -7 and ad + bc = 3 + 6 = 9, so 130 = 49 + 81 = 7² + 9². Choice A equals 109, choice B equals 128 and choice C equals 117.
Question 15 of 20 · Short Answer
Prove that (x + 4)(x² - 4x + 16) = x³ + 64.
Distribute the left side: x · x² - x · 4x + x · 16 + 4 · x² - 4 · 4x + 4 · 16 = x³ - 4x² + 16x + 4x² - 16x + 64. The x² terms cancel and the x terms cancel, leaving x³ + 64, the right side. This is the sum of cubes identity with b = 4.
Question 16 of 20 · Short Answer
Prove that (a + b)² - 2ab = a² + b². Then use it to find a² + b² when a + b = 9 and ab = 20, without finding a and b.
Expand: a² + 2ab + b² - 2ab = a² + b². Then a² + b² = 9² - 2(20) = 81 - 40 = 41. Check: the numbers 4 and 5 have sum 9 and product 20, and 16 + 25 = 41.
Question 17 of 20 · Short Answer
Give a counterexample showing that (x - y)² = x² - y² is not an identity. Then write the correct identity for (x - y)².
Example: x = 3, y = 1 gives (3 - 1)² = 4 on the left and 9 - 1 = 8 on the right, so the sides differ. The correct identity is (x - y)² = x² - 2xy + y². (Any values with y ≠ 0 and y ≠ x work as a counterexample.)
Question 18 of 20 · Short Answer
Use an identity to compute 998² without a calculator. Name the identity you used.
Write 998 = 1000 - 2 and use (a - b)² = a² - 2ab + b²: 1,000,000 - 4000 + 4 = 996,004.
Question 19 of 20 · Short Answer
Use the official identity with x = 5 and y = 3 to produce a Pythagorean triple. Is it primitive (no common factor), or a multiple of a smaller triple?
x² - y² = 16, 2xy = 30 and x² + y² = 34. Check: 256 + 900 = 1156 = 34². All three numbers are even, so the triple is not primitive: it is 2 times 8, 15, 17. This happens because x and y are both odd.
Question 20 of 20 · Short Answer
Prove that for every whole number n, the sum of the squares of n - 1, n and n + 1, minus 2, is 3 times a perfect square. Check with 4, 5 and 6.
HSA.APR.C.4 means students can prove that a polynomial equation is true for all values of its variables and can use such identities to explain facts about numbers. Typical identities are the difference of squares, the square of a sum, and the sum and difference of cubes. The official example uses (x² + y²)² = (x² - y²)² + (2xy)² to generate Pythagorean triples.
Is HSA.APR.C.4 taught in Algebra 1 or Algebra 2?
It is usually taught in Algebra II. Students meet the square of a sum and the difference of squares when factoring in Algebra I, but proving identities in general and using them to explain number patterns is usually part of the Algebra II polynomial unit.
What is the difference between an equation and an identity?
An equation can be true for only some values, while an identity is true for every value. The equation x² = 25 is true only for x = 5 and x = -5. The identity x² - 25 = (x - 5)(x + 5) is true for every number x. Solving an equation finds values; proving an identity shows that no value can fail.
Why is checking a few numbers not a proof of an identity?
Because a finite number of checks cannot rule out a value that fails. The expression n² + n + 41 gives a prime number for n = 0 through 39 and fails at n = 40. Checks are useful for finding counterexamples and for catching mistakes in a proof, but only an algebraic argument covers every value.
How do you write a proof of a polynomial identity?
Start from one side and rewrite it step by step until it becomes the other side. Each step should use a property of operations (distributive, commutative, associative) or an identity already proved. If one side is hard to work with, rewrite both sides until they reach the same expression. Avoid adding or subtracting terms on both sides, because that treats the identity as if it were already true.
How does the identity in the official example generate Pythagorean triples?
If you choose whole numbers x > y > 0, the three numbers x² - y², 2xy and x² + y² are whole numbers, and the identity says the square of the third equals the sum of the squares of the first two. So they are the side lengths of a right triangle. For example, x = 5 and y = 2 give 21, 20 and 29.
What mistakes do students make with polynomial identities?
A common mistake is to "distribute" an exponent over a sum, writing (a + b)² = a² + b² and forgetting the middle term 2ab. Other frequent errors are sign mistakes in (a - b)³, confusing the sum and difference of cubes factorizations, and treating a few successful numerical checks as a proof.
How are polynomial identities used outside of algebra class?
Identities give shortcuts and explanations for number facts. The difference of squares makes products like 48 · 52 quick to compute mentally, the square of a sum explains why 105² = 11,025, and identities like the one in the official example organize whole families of numbers such as Pythagorean triples. Later courses use identities to simplify expressions in calculus and in trigonometry.
How is HSA.APR.C.4 connected to other standards?
It builds on multiplying polynomials (HSA.APR.A.1) and seeing structure in expressions (HSA.SSE.A.2). The Binomial Theorem (HSA.APR.C.5) is the general identity for (x + y)ⁿ, and the (+) standard HSN.CN.C.8 extends identities to complex numbers, for example x² + 4 = (x + 2i)(x - 2i).
How can parents help with proving identities at home?
Ask your student to explain each line of a proof out loud: which rule turned this line into the next one? Number tricks are also a good way in: ask them to try "multiply two numbers that differ by 2 and add 1" with a few numbers, then ask why the answer is always a perfect square. The why is the identity n(n + 2) + 1 = (n + 1)².
07
Related Standards
6 standards
These standards connect to HSA.APR.C.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSA.APR.A.1Prerequisite
Add, subtract and multiply polynomials; polynomials are closed under these operations
