Perfect square trinomials: Level 2
Problem 8. Without multiplying out
a) explain why (1 − x)² = (x − 1)².
Because (1 − x) is the negative of (x − 1). And (−a)² = a² for any quantity a.
b) explain why (1 − x)³ = −(x − 1)³.
(−a)³ = −a³ for any quantity a.
*
The following problems show how we can go from what we know to what we do not know.
Problem 9. Use your knowledge of (a + b)² to multiply out (a + b)3.
[Hint: (a + b)3 = (a + b)(a + b)²]
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| (a + b)(a + b)² | = | (a + b)(a² + 2ab+ b²) | |
| = | a3 + 2a²b | + ab² | |
| + a²b | + 2ab² + b3 | ||
| (a + b)3 | = | a3 + 3a²b | + 3ab² + b3 |
Problem 10. Multiply out (x + 2)3.
| (x + 2)(x + 2)² | = | (x + 2)(x² + 4x+ 4) | |
| = | x3 + 4x² | + 4x | |
| + 2x² | + 8x + 8 | ||
| (x + 2)3 | = | x3 + 6x² | + 12x + 8 |
Problem 11. Multiply out (x − 1)3.
| (x − 1)(x − 1)² | = | (x− 1)(x² − 2x + 1) | |
| = | x3 − 2x² | + x | |
| − x² | + 2x − 1 | ||
| (x − 1)3 | = | x3 − 3x² | + 3x − 1 |
Problem 12. The square of a trinomial. Use your knowledge of
(a + b)² to multiply out (a + b + c)².
[Hint: Treat 
as a binomial with 
as the first term.]
Show that it will equal the sum of the squares of each term, plus twice the product of all combinations of the terms.
( + c)² |
= | (a + b)² + 2(a + b)c + c² |
| = | a² + 2ab + b² + 2ac + 2bc + c² | |
| = | a² + b² + c² + 2ab + 2ac + 2bc | |
Problem 13. Can you generalize the result of the previous problem? Can you immediately write down the square of (a + b + c + d)?
| (a + b + c + d)² = | a² | + b² + c² + d² |
| + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd | ||
Completing the square
x² + 8x + _?_ = (x + _?_)²
When the coefficient of x² is 1, as in this case, then to complete a perfect square trinomial, we must add a square number. What square number must we add?
We must add the square of half of coefficient of x. The trinomial will then be the square of x plus half that coefficient.
x² + 8x + 16 = (x + 4)²
We add the square of half the coefficient of x -- which in this case is 4 -- because when we multiply (x + 4)², the coefficient of x will be twice that number.
Problem 14.
| a) How do we indicate half of any number b? | b 2 |
| b) How do we indicate half of any fraction | p q |
? | p 2q |
(Skill in Arithmetic, Lesson 26.)
Example 7. Complete the square: x² − 7x + ? = (x − ?)²
| Solution. We will add the square of half of 7, which we write as | 7 2 |
. |
| x² − 7x + | 49 4 |
= | (x − | 7 2 |
)² |
And since the middle term of the trinomial has a minus sign, then the binomial also must have a minus sign.
Problem 15. Complete the square. The trinomial is the square of what binomial?
a) x² + 4x + ? x² + 4x + 4 = (x + 2)²
b) x² − 2x + ? x² − 2x + 1 = (x − 1)²
c) x² + 6x + ? x² + 6x + 9 = (x + 3)²
d) x² − 10x + ? x² − 10x + 25 = (x − 5)²
e) x² + 20x + ? x² + 20x + 100 = (x + 10)²
| f) x² + 5x + ? x² + 5x + | 25 4 |
= (x + | 5 2 |
)² |
| g) x² − 9x + ? x² − 9x + | 81 4 |
= (x − | 9 2 |
)² |
| h) x² + bx + ? x² + bx + | b² 4 |
= (x + | b 2 |
)² |
| i) x² + | b a |
x + ? | x² + | b a |
x + | b² 4a² |
= (x + | b 2a |
)² |
Next Lesson: The difference of two squares
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