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
Common Core (Algebra)
Common Core for Mathematics
Square of a Binomial
(a + b)2
= (a + b)(a + b)
= a (a + b) + b(a + b)
+ ab + ba + b2
+ 2ab + b2
(a − b)2
= (a − b)(a − b)
= a (a − b) − b(a − b)
− ab − ba + b2
− 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)
− ab + ba − b2
Sum and Difference of Two Cubes
Sum of Two Cubes
= (a + b)(a2
− ab + b2
Difference of Two Cubes
= (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.
Suppose that m and n are positive integers such that m > n.
Then the numbers m2
, 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= (m2
− n2)2+ (2mn)2
Expanding the left side, we get
(m2+ n2)2 = m4 + 2m2n2 + n4
Expanding the right side, we gat
(m2− n2)2+ (2mn)2
= m4 − 2m2n2 + n4 + 4m2n2
m4 + 2m2n2 + n4
Since the two expressions are identical, we have proven that
(m2+ n2)2= (m2− n2)2+ (2mn)2
Constructing Pythagorean Triples.
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