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Polynomial Identities


Videos and lessons to help High School students learn to 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.

Suggested Learning Targets

  • Understand that polynomial identities include but are not limited to the product of the sum and difference of two terms, the difference of two squares, the sum and difference of two cubes, the square of a binomial, etc .
  • Prove polynomial identities by showing steps and providing reasons.
  • Illustrate how polynomial identities are used to determine numerical relationships such as 252 = (20+5)2 = 202 + 2 • 20 • 5 + 52

Common Core: HSA-APR.C.4

Related Topics:
Common Core (Algebra)

Common Core for Mathematics

Square of a Binomial

(a + b)2
= (a + b)(a + b)
= a (a + b) + b(a + b)
= a2 + ab + ba + b2
= a2 + 2ab + b2

(a − b)2
= (a − b)(a − b)
= a (a − b) − b(a − b)
= a2 − ab − ba + b2
= a2 − 2ab + b2

Difference of Squares

Product of the sum and difference of two terms = Difference of Squares

(a + b)(a − b)
= a(a − b) + b(a − b)
= a2 − ab + ba − b2
= a2− b2

Sum and Difference of Two Cubes

Sum of Two Cubes
a3 + b3 = (a + b)(a2 − ab + b2)

Difference of Two Cubes
a3 − b3 = (a − b)(a2 + ab + b2)

Sum and Difference of 2 Cubes.

Understanding why the sum of two cubes is factored into its common formula representation.

A geometric interpretation of the Sum of Two Cubes formula.

A geometric interpretation of the Difference of Two Cubes formula.

Pythagorean Triple

Suppose that m and n are positive integers such that m > n.
Then the numbers m2+ n2, m2−n2, and 2mn are the lengths of the sides of a right triangle and form a Pythagorean Triple.

We can prove that by showing that
(m2+ n2)2= (m2n2)2+ (2mn)2

Expanding the left side, we get
(m2+ n2)2 = m4 + 2m2n2 + n4

Expanding the right side, we gat
(m2n2)2+ (2mn)2
= m4 − 2m2n2 + n4 + 4m2n2
= m4 + 2m2n2 + n4

Since the two expressions are identical, we have proven that
(m2+ n2)2= (m2n2)2+ (2mn)2

Constructing Pythagorean Triples.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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