Polynomial Identities
Related Topics:
Common Core (Algebra)
Common Core for Mathematics
Videos, solutions, examples, 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.
= (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
Finding the square of a binomial with one variable
Example:
Simplify (7x + 10)2
How to find the square of a binomial?
Example:
1. Expand each of the following:
(x + 5)2
(c - 7)2
(3d + 11)2
2. Which of the following are squares of binomials?
x2 + 49
x2 + 6x + 9
4c2 - 20c - 25
p2 - 30p + 225
9y2 - 24y + 16
4f2 + 10f + 25
(a + b)(a − b)
= a(a − b) + b(a − b)
= a2− ab + ba − b2
= a2− b2
How to multiply the sum and difference of two terms?
Example:
Multiply:
(4 - 3y)(4 + 3y)
(2x + 7)(2x - 7)
(9x + 5)(9x - 5)
(y2 - 2)(y2 + 2)
How to find the difference of squares?
Factor x2 - 9
Factor x2 - 16
Factor 15x2 - 144
Factor a2 - b2
Solve the equation x2 = 100
a3 + b3 = (a + b)(a2 − ab + b2)
Difference of Two Cubes
a3 − b3 = (a − b)(a2 + ab + b2)
How to find the 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
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 = (m2 - n2)2 + (2mn)2
Expanding the left side, we get
(m2 + n2)2 = m4 + 2m2n2 + n4
Expanding the right side, we get
(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
Common Core (Algebra)
Common Core for Mathematics
Videos, solutions, examples, 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
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
Finding the square of a binomial with one variable
Example:
Simplify (7x + 10)2
How to find the square of a binomial?
Example:
1. Expand each of the following:
(x + 5)2
(c - 7)2
(3d + 11)2
2. Which of the following are squares of binomials?
x2 + 49
x2 + 6x + 9
4c2 - 20c - 25
p2 - 30p + 225
9y2 - 24y + 16
4f2 + 10f + 25
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
How to multiply the sum and difference of two terms?
Example:
Multiply:
(4 - 3y)(4 + 3y)
(2x + 7)(2x - 7)
(9x + 5)(9x - 5)
(y2 - 2)(y2 + 2)
How to find the difference of squares?
Factor x2 - 9
Factor x2 - 16
Factor 15x2 - 144
Factor a2 - b2
Solve the equation x2 = 100
Sum and Difference of Two Cubes
Sum of Two Cubesa3 + b3 = (a + b)(a2 − ab + b2)
Difference of Two Cubes
a3 − b3 = (a − b)(a2 + ab + b2)
How to find the 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 = (m2 - n2)2 + (2mn)2
Expanding the left side, we get
(m2 + n2)2 = m4 + 2m2n2 + n4
Expanding the right side, we get
(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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