18
THE SQUARE OF A BINOMIAL
Perfect square trinomials
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
(x + 5)²
The square of a binomial come 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 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. To see what happens when we square any binomial, let us square (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 an important product 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· 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 = 9x² − 24x + 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 a − b, then the middle term will be −2ab
Example 3. (5x3 − 1)² = 25x6 − 10x3 + 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) (x − y)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)²
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 )² | f) | 16m² − 40mn+ 25n² = (4m − 5n)² | |
| g) | x4 + 2x²y² + y4 = (x² + y²)² | h) | 4x6 − 10x3y4 + 25y8 Not possible. | |
Geometrical algebra
Here is a square whose side is a + b.
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.
Next Lesson: The difference of two squares
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