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18

THE SQUARE OF A BINOMIAL

Perfect square trinomials

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

(a + b)2

The square of a binomial comes up so often that the student should be able to write the final product immediately.  It will turn out to be a very specific trinomial.  To see that, let us square (a + b):

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

For, the outers plus the inners will be

ab + ba = 2ab.

The square of any binomial produces the following trinomial:

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

These will be the 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 a binomial is a essential form in the "multiplication table" of algebra.

See Lesson 8 of Arithmetic: How to square a number mentally, particularly the square of 24, which is the "binomial" 20 + 4.

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 with 6.  (x · 6 = 6x.  Twice that is 12x.)

36 is the square of 6.

The square of a binomial is called a perfect square trinomial.

x2 + 12x + 36 is a perfect square trinomial.

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

The square of +b or −b, of course, is always positive.  It is always +b2.

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

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

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

1 is the square of −1.

The student should be clear that (a + b)2 is not a2 + b2, any more than (a + b)3 is equal to a3 + b3.

An exponent may not be "distributed" over a sum.

(See Topic 25 of Precalculus: The binomial theorem.)

Problem 1.

a)  State in words the rule 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 2.   Write only the trinomial product.

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

Problem 3.   Write only the trinomial product.

   a)   (2x + 1)24x2 + 4x + 1   b)  (3x − 2)2 =  9x2 − 12x + 4
 
   c)   (4x + 3)216x2 + 24x + 9   d)  (5x − 2)2 =  25x2 − 20x + 4
 
   e)   (x3 + 1)2x6 + 2x3 + 1   f)  (x4 − 3)2 =  x8 − 6x4 + 9
 
   g)   (xn + 1)2x2n + 2xn + 1   h)  (xn − 4)2 =  x2n − 8xn + 16

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 4.   Factor:  p2 + 2pq + q2.

p2 + 2pq + q2 = (p + q)2

The left-hand side is a perfect square trinomial.

Problem 5.   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 6.   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 )2

 f)   16m2 − 40mn+ 25n2 = (4m − 5n)2

 g)   x4 + 2x2y2 + y4 = (x2 + y2)2

 h)   4x6 − 10x3y4 + 25y8 Not possible.

 i)   x12 + 8x6 + 16 = (x6 + 4)2

 j)   x2n + 8xn + 16 = (xn + 4)2

Geometrical algebra

Here is a square whose side is a + b.

A square

It is composed of

a square whose side is a,

a square whose side is b,

and two rectangles ab.

That is,

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

2nd Level

end

Next Lesson:  The difference of two squares

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