Algebra (Period 1)

Trinomial Squares 6-3

This is a special product that we learned in Chapter 5 when we did FOILing.

FOIL:
(a + 3)² (called a binomial squared)

(a + 3)(a + 3) = a² + 3a + 3a + 9
a² + 6a + 9 (called a trinomial square)



Again, you see that the middle term is DOUBLE the product of the two terms in the binomial, and the first and last terms are simply the squares of each term in the binomial.
   
HOW TO RECOGNIZE THAT IT IS A BINOMIAL SQUARED:

1) Is it a trinomial? (if it's a binomial, it cannot be a binomial squared - it may be diff of 2 squares)

2) Are the first and last terms POSITIVE?
3) Are the first and last terms perfect squares?

4) Is the middle term double the product of the square roots of the first and last terms?


IF YES TO ALL OF THESE QUESTIONS, THEN YOU HAVE A TRINOMIAL SQUARE
 
TO FACTOR A TRINOMIAL SQUARE: a²
 + 6a + 9
1) (     )²

2) Put the sign of the middle term in the (   +   )²

3) Find the square root of the first term and the last term and place in the parentheses:  
(3 + a)²

4) Check by FOILing back. (or using the BOX method)

x² - 14x + 49
(x -7)²

16x² - 56xy + 49y²
ask yourself those important questions. They are all YES... so
(4x - 7y)²

x² -4xy + 4y²
(x - 2y)²
y⁶ + 16y³ + 64
(y³ + 8)²

9a⁸ - 30a⁴b + 25b²
(3a⁴ -5b)²

What about
2x² -40x + 200
What's the first thing you must look for? ALWAYS!!! the GCF
2(x² -20x + 100) now it is a trinomial square
2(x -10)²

similarly with
2x² -4x + 2
2(x² -2x + 1)... again NOW it is a trinomial square
2(x -1)²

18x³ + 12x² + 2x
What's the GCF? 2x

2x(9x² + 6x + 1)
2x(3x + 1)²

But look at
(a +4)² - 2(a +4) + 1
How can that be a trinomial square?
Well, think about the following
A² + 2AB + B² and
A² - 2AB + B²
represent generic trinomial squares
so if we let (a + 4) represent A... it works
[(a +4) -1]²
but we can simplify that to
(a +3)²

You could multiply everything out and then combine like terms,
that is,
(a +4)² - 2(a +4) + 1 becomes
a² -8a + 16 -2a -8 + 1 which then simplifies
a² -6a + 9 which is definitely a trinomial square
(a-3)²
but that's what we got using a simpler method!!

Try
(y+3)² + 2(y+3) + 1
It is a trinomial square in the form
A² + 2AB + B²
so [(y + 3) +1]²
which becomes (y +4)²