Take Note Theorem: The Fundamental Theorem of Algebra

If P(x) is a polynomial of degree n ≥ 1 , n greater than or equal to 1 comma  then P ( x ) = 0 p open x close equals 0  has exactly n roots, including multiple and complex roots.

Problem 1 Using the Fundamental Theorem of Algebra

What are all the roots of x 5 − x 4 − 3 x 3 + 3 x 2 − 4 x + 4 = 0 ? x to the fifth , minus , x to the fourth , minus , 3 x cubed , plus , 3 x squared , minus 4 x plus 4 equals 0 question mark

• Step 1 The polynomial is in standard form. The possible rational roots are ± 1 , ± 2 , ± 4 . plus minus 1 comma plus minus 2 comma plus minus 4 .

• Step 2 Evaluate the related polynomial function for x = 1 . x equals 1 .  Since P ( 1 ) = 0 , 1 p open 1 close equals 0 comma 1  is a root and x − 1 x minus 1  is a factor. Use synthetic division to factor out x − 1 : x minus 1 colon

  1 1 − 1 − 3 3 − 4 4     1 0 − 3 0 − 4   1 0 − 3 0 − 4 0 ¯ table with 3 rows and 1 column , row1 column 1 , table with 1 row and 7 columns , row1 column 1 , 1 , column 2 1 , column 3 negative 1 , column 4 negative 3 , column 5 3 , column 6 negative 4 , column 7 4 , end table , row2 column 1 , table with 1 row and 7 columns , row1 column 1 , , column 2 , column 3 1 , column 4 0 , column 5 negative 3 , column 6 0 , column 7 negative 4 , end table , row3 column 1 , table with 1 row and 7 columns , row1 column 1 , , column 2 1 , column 3 0 , column 4 negative 3 , column 5 0 , column 6 negative 4 , column 7 0 , end table bar , end table

• Step 3 Continue factoring until you have five linear factors.

  x 5 − x 4 − 3 x 3 + 3 x 2 − 4 x + 4 = ( x − 1 ) ( x 4 − 3 x 2 − 4 )   = ( x − 1 ) ( x 2 − 4 ) ( x 2 + 1 )   = ( x − 1 ) ( x − 2 ) ( x + 2 ) ( x − i ) ( x + i ) table with 3 rows and 2 columns , row1 column 1 , x to the fifth , minus , x to the fourth , minus 3 , x cubed , plus 3 , x squared , minus 4 x plus 4 , column 2 equals . open , x minus 1 , close . open . x to the fourth , minus 3 , x squared , minus 4 . close , row2 column 1 , , column 2 equals . open , x minus 1 , close . open . x squared , minus 4 . close . open . x squared , plus 1 . close , row3 column 1 , , column 2 equals . open , x minus 1 , close . open , x minus 2 , close . open , x plus 2 , close . open , x minus i , close . open , x plus i , close , end table

• Step 4 The roots are 1 , 2 , − 2 , i ,  and  − i . 1 comma 2 comma negative 2 comma i comma , and , minus i .

  By the Fundamental Theorem of Algebra, these are the only roots.

✓ Got It?

1. What are all the roots of the equation x 4 + 2 x 3 = 13 x 2 − 10 x ? x to the fourth , plus 2 , x cubed , equals , 13 x squared , minus 10 x question mark