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.

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For example, the polynomial identity (x*^{2} + y^{2})^{2} = (x^{2} – y^{2})^{2} + (2xy)^{2} can be used to generate Pythagorean triples.

### Suggested Learning Targets

### Square of a Binomial

(a + b)^{2}

= (a + b)(a + b)

= a (a + b) + b(a + b)

= a^{2} + ab + ba + b^{2}

= a^{2} + 2ab + b^{2}

(a − b)^{2}

= (a − b)(a − b)

= a (a − b) − b(a − b)

= a^{2} − ab − ba + b^{2}

= a^{2} − 2ab + b^{2}

**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?

x^{2} + 49

x^{2} + 6x + 9

4c^{2} - 20c - 25

p^{2} - 30p + 225

9y^{2} - 24y + 16

4f^{2} + 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)

= a^{2}− ab + ba − b^{2}

= a^{2}− b^{2}

**How to multiply the sum and difference of two terms?**

Example:

Multiply:

(4 - 3y)(4 + 3y)

(2x + 7)(2x - 7)

(9x + 5)(9x - 5)

(y^{2} - 2)(y^{2} + 2)

**How to find the difference of squares?**

Factor x^{2} - 9

Factor x^{2} - 16

Factor 15x^{2} - 144

Factor a^{2} - b^{2}

Solve the equation x^{2} = 100

### Sum and Difference of Two Cubes

Sum of Two Cubes

a^{3} + b^{3} = (a + b)(a^{2} − ab + b^{2})

Difference of Two Cubes

a^{3} − b^{3} = (a − b)(a^{2} + ab + b^{2})

**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 m^{2}+ n^{2}, m^{2}−n^{2}, and 2mn are the lengths of the sides of a right triangle and form a Pythagorean Triple.

We can prove that by showing that

(m^{2} + n^{2})^{2} = (m^{2} - n^{2})^{2} + (2mn)^{2}

Expanding the left side, we get

(m^{2} + n^{2})^{2} = m^{4} + 2m^{2}n^{2} + n^{4}

Expanding the right side, we get

(m^{2} - n^{2})^{2} + (2mn)^{2}

= m^{4} − 2m^{2}n^{2} + n^{4}^{} + 4m^{2}n^{2}

= m^{4} + 2m^{2}n^{2} + n^{4}^{}

Since the two expressions are identical, we have proven that

(m^{2} + n^{2})^{2} = (m^{2} - n^{2})^{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.

- 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
25
^{2}= (20+5)^{2}= 20^{2}+ 2 • 20 • 5 + 5^{2}

= (a + b)(a + b)

= a (a + b) + b(a + b)

= a

= a

(a − b)

= (a − b)(a − b)

= a (a − b) − b(a − b)

= a

= a

Example:

Simplify (7x + 10)

Example:

1. Expand each of the following:

(x + 5)

(c - 7)

(3d + 11)

2. Which of the following are squares of binomials?

x

x

4c

p

9y

4f

(a + b)(a − b)

= a(a − b) + b(a − b)

= a

= a

Example:

Multiply:

(4 - 3y)(4 + 3y)

(2x + 7)(2x - 7)

(9x + 5)(9x - 5)

(y

Factor x

Factor x

Factor 15x

Factor a

Solve the equation x

a

Difference of Two Cubes

a

Then the numbers m

We can prove that by showing that

(m

Expanding the left side, we get

(m

Expanding the right side, we get

(m

= m

= m

Since the two expressions are identical, we have proven that

(m

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