Algebra Demystified 2nd Ed

160 algebra De mystif ieD

✔SOLUTIONS

1. x^2 − 4 = (x − 2)(x + 2)


2. x^2 − 81 = (x − 9)(x + 9)


3. x^2 − 25 = (x − 5)(x + 5)


4. x^2 − 64 = (x − 8)(x + 8)


5. x^2 − 1 = (x − 1)(x + 1)


6. xx^2 –  15   = ()–  15 ()x +  15


7. 25 − x^2 = (5 − x)(5 + x)

The difference of two squares can come in the form xc^22 nn− , where n is any

nonzero number. The factorization is xc^22 nn−=()xcnn−+()xcnn. (When n is odd,

we can factor xcnn− as well, but this factorization will not be covered here.) In

the following problems, we make use of the fact that 1n = 1 for any exponent n.

EXAMPLES

Use the formula to factor the difference of two squares.

x^6 – 1

The powers are 2 ⋅ 3 = 6. Thus,

xx^66 –  11   =  – ^62 = xx()^32 –  11 ()^33  (=  – ^33 )(x +  13 )) = ()()xx^33 –  11 – .

x^10 – 1 = x^10 – 1^10 = (x^5 – 1)(x^5 + 1)

xx^632 xx

2

(^133)
64
1
8
1
8
1
8
–   )= (– =  – +
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Sometimes, we need to use the formula twice.
16 – x^4 = 2^4 – x^4 = (2^2 – x^2 )(2^2 + x^2 ) = (4 – x^2 )(4 + x^2 )
Notice that the factor 4  – x^2 is also the difference of two squares, so it, too,
can be factored, as in (4 – x^2 )(4 + x^2 ) = (2 – x)(2 + x)(4 + x^2 ).
16 x^4 – 1 = (2x)^4 – 1^4 = (4x^2 – 1)(4x^2 + 1) = (2x – 1)(2x + 1)(4x^2 + 1)
x^8 – 1 = x^8 – 1^8 =(x^4 – 1)(x^4 + 1) = (x^2 – 1)(x^2 + 1)(x^4 + 1)
= (x – 1)(x + 1)(x^2 + 1)(x^4 + 1)
EXAMPLES
Use the formula to factor the difference of two squares.