2.1. Factoring Review http://www.ck12.org
=^13 x(x^4 − 11 x^2 + 18 )
=^13 x(x^2 − 2 )(x^2 − 9 )
=^13 x(x+√
2 )(x−√
2 )(x+ 3 )(x− 3 )- −^27 x^4 +^7463 x^2 − 638. Letu=x^2.
=−^27 u^2 +^7463 u− 638
=−^27(
u^2 −^379 u+^49)
Factoring through fractions like this can be extremely tricky. You must recognize that−^19 and -4 sum to−^379 and
multiply to^49.
=−^27
(
u−^19)
(u− 4 )=−^27(
x^2 −^19)
(x^2 − 4 )=−^27(
x−^13)(
x+^13)
(x− 2 )(x+ 2 )3.x^4 +x^2 − 72
= (x^2 − 8 )(x^2 + 9 )
Notice that(x^2 − 8 )can be written as the difference of perfect squares because 8=
(√
8
) 2
=
(
2
√
2
) 2
. On the
other hand,x^2 +9 cannot be written as the difference between squares because thex^2 and the 9 are being added not
subtracted. This polynomial cannot be factored into strictly linear factors.
= (x− 2
√
2 )(x+ 2√
2 )(x^2 + 9 )Practice
Factor each polynomial into strictly linear factors if possible. If not possible, explain why not.
1.x^2 + 5 x+ 6
2.x^4 + 5 x^2 + 6
3.x^4 − 16- 2x^2 − 20
- 3x^2 + 9 x+ 6
- x 24 − 5 x^2 +^92
7.^2 x 34 −^343 x^2 +^323
8.x^2 −^14
9.x^4 −^374 x^2 +^94
10.^34 x^4 −^874 x^2 + 75