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18

PERFECT SQUARE
TRINOMIALS

The square of a binomial

The square numbers

2nd level

(a + b


LET US BEGIN by learning about the square numbers.  They are the numbers

1· 1   2· 2   3· 3

and so on.  The following are the first ten square numbers -- and their roots.

Square numbers 1 4 9 16 25 36 49 64 81 100
Square roots 1 2 3 4 5 6 7 8 9 10

1 is the square of 1.   4 is the square of 2.   9 is the square of 3.  And so on.

The square root of 1 is 1.  The square root of 4 is 2.   The square root of 9 is 3.  And so on.

In a multiplication table, the square numbers lie along the diagonal.


The square of a binomial

Let us square the binomial (x + 5):

(x + 5)² = (x + 5)(x + 5) = x² + 10x + 25

(See Lesson 16:  Quadratic trinomials.)

x² + 10x + 25 is called a perfect square trinomial.  It is the square of a binomial.

The square of a binomial come up so often that the student should be able to write the trinomial product quickly and easily.  Therefore, let us see what happens when we square any binomial, a + b :

(a + b)² = (a + b)(a + b) = a² + 2ab + b²

The square of any binomial produces the following three terms:

1.   The square of the first term of the binomial:  a²

2.   Twice the product of the two terms:  2ab

3.   The square of the second term:  b²

The square of every binomial -- every perfect square trinomial -- has that form:  a² + 2ab + b².  To recognize that is to know the "multiplication table" of algebra.

Example 1.   Square the binomial (x + 6).

Solution.    (x + 6)² = x² + 12x + 36

x² is the square of x.

12x  is  twice the product of  x· 6.  (x· 6 = 6x.  Twice that is 12x.)

36 is the square of 6.

Example 2.   Square the binomial (3x − 4).

Solution.    (3x − 4)² = 9x²24x + 16

9x² is the square of 3x.

−24x  is  twice the product of  3x· −4.  (3x· −4 = −12x.  Twice that is −24x.)

16 is the square of −4.

Note:  If the binomial has a minus sign, then the minus sign appears only in the middle term of the trinomial.   Therefore, using the double sign  ±  ("plus or minus"), we can state the rule as follows:


(a ± b)² = a² ± 2ab + b²

This means:  If the binomial is a + b, then the middle term will be +2ab;  but if the binomial is ab, then the middle term will be −2ab

Example 3.   (5x3 − 1)² = 25x610x3 + 1

25x6 is the square of 5x3.  (Lesson 13:  Exponents.)

−10x3  is  twice the product of  5x3· −1.  (5x3· −1 = −5x3.  Twice that is −10x3.)

1 is the square of −1.

Example 4.   Is this a perfect square trinomial:  x² + 14x + 49 ?

Answer.   Yes.  It is the square of (x + 7).

x² is the square of x.  49 is the square of 7.  And 14x is twice the product of x· 7.

In other words, x² + 14x + 49 could be factored as

x² + 14x + 49 = (x + 7)²

Note:  If the coefficient of x had been any number but 14, this would not have been a perfect square trinomial.

Example 5   Is this a perfect square trinomial:  x² + 50x + 100 ?

Answer.   No, it is not.   Although x² is the square of x, and 100 is the square of 10,  50x is not twice the product of x· 10.  (Twice their product is 20x.)

Example 6   Is this a perfect square trinomial:  x8 − 16x4 + 64 ?

Answer.   Yes.  It is the perfect square of  x4 − 8.

Problem 1.   Which numbers are the square numbers?

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Do the problem yourself first!

1,  4,  9,  16,  25,  36,  49,  64,  etc.

These are the numbers  1²,  2²,  3²,  and so on.

Problem 2.

a)  State in words the formula for squaring a binomial.

The square of the first term.
Twice the product of the two terms.
The square of the second term.

b)  Write only the trinomial product:  (x + 8)² =  x² + 16x + 64

c)  Write only the trinomial product:  (r + s)² =  r² + 2rs + s²

Problem 3.   Write only the trinomial product.

   a)   (x + 1)² = x² + 2x + 1   b)  (x − 1)² =  x² − 2x + 1
 
   c)   (x + 2)² = x² + 4x + 4   d)  (x − 3)² =  x² − 6x + 9
 
   e)   (x + 4)² = x² + 8x + 16   f)  (x − 5)² =  x² − 10x + 25
 
   g)   (x + 6)² = x² + 12x + 36   h)  (xy)² =  x² − 2xy + y²

Problem 4.   Write only the trinomial product.

   a)   (2x + 1)² = 4x² + 4x + 1   b)  (3x − 2)² =  9x² − 12x + 4
 
   c)   (4x + 3)² = 16x² + 24x + 9   d)  (5x − 2)² =  25x² − 20x + 4
 
   e)   (x3 + 1)² = x6 + 2x3 + 1   f)  (x4 − 3)² =  x8 − 6x4 + 9
 
   g)   (xn + 1)² = x2n + 2xn + 1   h)  (xn − 4)² =  x2n − 8xn + 16

Problem 5.   Factor:  p² + 2pq + q².

p² + 2pq + q² = (p + q
The left-hand side is a perfect square trinomial.

Problem 6.   Factor as a perfect square trinomial -- if possible.

   a)   x² − 4x + 4 = (x − 2)²   b)   x² + 6x + 9 = (x + 3)²
 
   c)   x² − 18x + 36  Not possible.   d)   x² − 12x + 36 = (x − 6)²
 
   e)   x² − 3x + 9  Not possible.   f)   x² + 10x + 25 = (x + 5)²

Problem 7.   Factor as a perfect square trinomial, if possible.

   a)   25x² + 30x + 9 = (5x + 3)²   b)   4x² − 28x + 49 = (2x − 7)²
 
   c)   25x² − 10x + 4  Not possible.   d)   25x² − 20x + 4 = (5x − 2)²
 
   e)   1 − 16y + 64y² = (1 − 8y   f)   16m² − 40mn+ 25n² = (4m − 5n
 
   g)   x4 + 2x²y² + y4 = (x² + y²)²   h)   4x6 − 10x3y4 + 25y8 Not possible.

2nd Level


Next Lesson:  The difference of two squares


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