http://www.ck12.org Chapter 2. Polynomials and Rational Functions
−^12 (u^2 − 7 u+ 12 ) =−^12 (u− 3 )(u− 4 )
=−^12 (x^2 − 3 )(x^2 − 4 )This type of temporary substitution that enables you to see the underlying structure of an expression is very common
in calculus.
Example C
Factor the resulting expression from Example B into four linear factors and a constant.
−^12 (x^2 − 3 )(x^2 − 4 )
Solution:Many students may recognize thatx^2 −4 immediately factors by the difference of squares method to be
(x− 2 )(x+ 2 ). This problem asks for more because sometimes the difference of squares method can be applied to
expressions likex^2 −3 where each term is not a perfect square. The number 3 actually is a square.
3 =
(√
3
) 2
So the expression may be factored to be:
−^12
(
x−√ 3)(
x+√ 3)
(x− 2 )(x+ 2 )Concept Problem Revisited
One reason why it might be useful to completely factor an expression like−^12 (x^4 − 7 x^2 + 12 )into linear factors is if
you wanted to find the roots of the functionf(x) =−^12 (x^4 − 7 x^2 + 12 ). The roots arex=±
√
3 ,±2.
You should recognize thatx^2 −3 can still be thought of as the difference of perfect squares because the number 3
can be expressed as
(√
3
) 2
. Rewriting the number 3 to fit a factoring pattern that you already know is an example
of using the basic factoring techniques at a PreCalculus level.
Vocabulary
Apolynomialis a mathematical expression that is often represented as a sum of terms or a product of factors.
To factormeans to rewrite a polynomial expression given as a sum of terms into a product of factors.
Linear factorsare expressions of the formax+bwhereaandbare real numbers.
Guided Practice
- Factor the following expression into strictly linear factors if possible. If not possible, explain why.
x 35 − 113 x^3 + 6 x - Factor the following expression into strictly linear factors if possible. If not possible, explain why.
−^27 x^4 +^7463 x^2 − 638 - Factor the following expression into strictly linear factors if possible. If not possible, explain why.
x^4 +x^2 − 72
Answers:
1.x 35 −^113 x^3 + 6 x