Special Products Difference of Two Squares Perfect Square Trinomials

Special ProductsDifference of Two SquaresPerfect Square Trinomials

Bell Ringer:Factor.

4x3 + 4x2 – 24x4x (x2 + x – 6)

4x (x + 3)(x – 2)

LEQs:How do you factor differences of squares?

How do you factor perfect square trinomials?

Perfect Square Trinomials

Perfect Square Trinomialsa2 + 2ab + b2 = (a + b)2

a2 – 2ab + b2 = (a b)2

In order for a polynomial to be a perfect squaretrinomial, two conditions must be satisfied:

1. The first and last terms must be perfect squares.

2. The “middle term” must equal 2 or – 2 times the product of the expressions being squared in the first and last term.

Perfect Square Trinomial Form: Example:

 (Form 1)a2 + 2ab + b2 = (a + b)2 x2 + 6x+ 9 = ____________ = (x + 3)2

a = _____ , b

= _____ , 2ab = _______

  (Form 2)a2 – 2ab + b2 = (a b)2 x2 – 10x + 25 = ____________ = (x 5)2

a = _____ , b = _____ , 2ab = _______  

Condition 1:First and last term are

perfect squares.

Condition 1:First and last term are

perfect squares.

Perfect Square Trinomial Form: Example:

 (Form 1)a2 + 2ab + b2 = (a + b)2 x2 + 6x+ 9 = ____________ = (x + 3)2

a = _____ , b

= _____ , 2ab = _______

  (Form 2)a2 – 2ab + b2 = (a b)2 x2 – 10x + 25 = ____________ = (x 5)2

a = _____ , b = _____ , 2ab = _______  

Condition 2:

The “middle term” must equal 2 or – 2 times the product of the expressions

being squared in the first and last term.

Condition 2:

The “middle term” must equal 2 or – 2 times the product of the expressions

being squared in the first and last term.

Perfect Square Trinomial Form: Example:

 (Form 1)a2 + 2ab + b2 = (a + b)2 x2 + 6x+ 9 = ____________ = (x + 3)2

a = _____ , b

= _____ , 2ab = _______

  (Form 2)a2 – 2ab + b2 = (a b)2 x2 – 10x + 25 = ____________ = (x 5)2

a = _____ , b = _____ , 2ab = _______  

Perfect Square Trinomials

Examples: Factor.

x2 + 8x + 16 = ____________ = (x + 4)2

1.) x2 + 8x + 16The first term, x2, and third term, 16, are perfect squares.

The middle term, 8x, is 2 times the product of x and 4.

2.) x2 – 14x + 49

Perfect Square Trinomials

3.) 9x4 30x2z + 25z2

Perfect Square Trinomials

4. 100a2 – 140ab + 49b2

Perfect Square Trinomials

Team Huddle

Team Mastery

Difference of Two SquaresDifference of Squares Form: Example:a2 – b2 = (a + b)(a b) x2 – 64 = ( ____ ____ ) ( ____ ____ )

Check using FOIL:

    Why do we call these DIFFERENCES What happens to the two

middle of two squares? terms when we FOIL a

difference of two squares?

Example 1: Factor x2 – 4

Notice the terms are both perfect squares

x2 = (x)2 4 = (2)2

x2 – 4 = (x)2 – (2)2

a2 – b2

and we have a difference

= (x – 2)(x + 2)

difference of squares

= (a – b)(a + b)factors as

Difference of Two Squares

Example 2: Factor 9p2 – 16

Notice the terms are both perfect squares

9p2 = (3p)2 16 = (4)2

9a2 – 16 = (3p)2 – (4)2

a2 – b2

and we have a difference

= (3p – 4)(3p + 4)

difference of squares

= (a – b)(a + b)factors as

Difference of Two Squares

Example 3: Factor 2y6 – 50

Difference of Two Squares

GCF First 2(y6 – 25)

Now it’s a difference of

squares!

Example 3: Factor 2y6 – 50

Notice the terms are both perfect squares

y6 = (y3)2 25 = (5)2

2(y6 – 25) = 2 ((y3)2 – (5)2)

2(a2 – b2)

and we have a difference

= 2(y3 – 5)(y3 + 5)

difference of squares

= 2(a – b)(a + b)factors as

Difference of Two Squares

GCF First 2(y6 – 25)

Example 4: Factor 81 – x2y2

Notice the terms are both perfect squares

81 = (9)2 x2y2 = (xy)2

81 – x2y2 = (9)2 – (xy)2

a2 – b2

and we have a difference

= (9 – xy)(9 + xy)

difference of squares

= (a – b)(a + b)factors as

Difference of Two Squares

Team Huddle

Team Mastery

IXL Math Practice

Algebra 1AA.5 Factor Quadratics: special cases

Quick Check:Factor:

1. x2-49 2. x2 + 16x + 64

Groups:

Group 1 – Work on HW IndividuallyGroup 2 – Work on HW with me!