Algebra Honors ( Period 6 & 7)

Squares of Binomials 5-6

(a +b)² =(a+b)(a+b)

You could use foil, fireworks, the box method… to find the results of multiplying the two binomials

a² + 2ab + b²

What happens when you square the binomial difference a-b?

(a-b)² = (a-b)(a-b)

a² - 2ab + b²

Notice you have the square of the first term,  twice the product of the two terms and finally the square of the last term.

The textbook states that it is helpful to memorize these patterns for writing squares of binomials as trinomials.

(a +b)²     = a² + 2ab + b²

(a - b)²     = a² -2ab + b²

My comment-  MEMORIZE … you need to be able to see these patterns

(x + 3)² = x² + 6x + 9

( 7u -3)² = 49u² -42u+ 9 

(4s – 5t)² = 16s² – 40st + 25t²

(3p² – 2q²)²

=9p⁴-12p²q² + 4q⁴

When we need to factor… we need to realize these patterns in reverse…that is,

a² + 2ab + b^{2  =}    (a +b)²     

a² -2ab + b²   =  (a - b)²     

a² + 2ab + b²     and  a² -2ab + b²       are called perfect square trinomials

because each expression has a three terms and is the square of a binomial.

To test whether a trinomial is a perfect square… ask these three questions:

1)  Is the first term a square?

2) Is the last term a square?

3) Is the middle term twice the product of √( 1^st term)  and √(last term).

For example, is

4x² – 20x + 25 a perfect square trinomial?

 1)  Is the first term a square? YES 4x² = (2x)²

2) Is the last term a square? YES  25= (5)²

3) Is the middle term twice the product of √( 1^st term)  and √(last term).

YES  2[√( 4x²) ∙ √25)]. =  2(2x∙5) = 20x

Yu may need to rearrange the terms of a trinomial BEFORE you test whether it is a perfect square.

For example, x² + 100 -20x must be rewritten as

x² -20x + 100  so that we can answer YES to all three questions.

Take a look at

63n³ – 84n² + 28n

It doesn’t look like a perfect square BUT if you factor out the GCF or the Greatest Monomial factor…

we end up with à 7n(9n² -12n +4)

Now… that becomes  7n(3n-2)²

What about,

8u3 -24u2v + 18uv2

2u(4u² -12vu +9v²)

2u(2u-3v)²

What do we need to do to solve the following:
(x + 2)² – (x -3) ² = 35

First multiply each of the binomial squares

x² + 4x + 4 – (x² -6x+ 9) = 35

Make sure to properly employ the inverse property of a sum

x² + 4x + 4 – x² + 6x -  9 = 35

Combine like terms

10x – 5 = 35

10x = 40

x = 4

or using set notation {4}