Algebra II

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 x ³ – x ² – 10 x – 8? 

Check to see whether ( x ³ – x ² – 10 x – 8) ÷ ( x + 2) has a remainder of zero. Using synthetic division, you get

Because the remainder of the division is zero, ( x + 2) is a factor of x ³ – x ² – 10 x – 8. The expression x ³ – x ² – 10 x – 8 can now be expressed in factored form. 

x ³ – x ² – 10 x – 8 = ( x + 2)( x ² – 3 x – 4) 

But ( x ² – 3 x – 4) can be factored further into ( x – 4)( x + 1). So 

x ³ – x ² – 10 x – 8 = ( x + 2)( x – 4)( x + 1) 

The expression x ³ – x ² – 10 x – 8 is now completely factored. From this form, it also is seen that ( x – 4) and ( x + 1) are also factors of x ³ – x ² – 10 x – 8.