Linear Sentences in One Variable
Polynomial Functions
If P( x) is a polynomial, then P( r)=0 if and only if x − r is a factor of P( x).
Example 1: Is ( x + 2) a factor of x3 − x2 − 10 x − 8?
Check to see if (
x3 −
x2 − 10
x − 8) ÷ (
x + 2) has a remainder of zero. Using synthetic division, you get
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Because the remainder of the division is zero, ( x + 2) is a factor of x3 − x2 − 10 x − 8. The expression x3 − x2 − 10 x − 8 can now be expressed in factored form.
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But (
x2 − 3
x − 4) can be factored further into (
x − 4)(
x + 1). So
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The expression x3 − x2 − 10 x − 8 is now completely factored. From this form, it is also seen that ( x − 4) and ( x + 1) are also factors of x3 − x2 − 10 x − 8.
