Problem 3 Using Synthetic Division

Use synthetic division to divide x 3 − 14 x 2 + 51 x − 54 x cubed , minus , 14 x squared , plus 51 x minus 54  by x + 2 . x plus 2 .  What is the quotient and remainder?

• Step 1 Reverse the sign of + 2 . plus 2 .  Write the coefficients of the polynomial.

  − 2 1 − 14 51 − 54 negative 2 , 1 minus 14 51 minus 54

• Step 2 Bring down the first coefficient.

  − 2 1 − 14 51 − 54 1 ¯ table with 2 rows and 1 column , row1 column 1 , negative 2 , 1 minus 14 51 minus 54 , row2 column 1 , 1 bar , end table

• Step 3 Multiply the coefficient by the divisor. Add to the next coefficient.

  

• Step 4 Continue multiplying and adding through the last coefficient.

  − 2 1 − 14 51 − 54 − 2 32 − 166 1 − 16 83 − 220 ¯ table with 3 rows and 1 column , row1 column 1 , negative 2 , 1 minus 14 51 minus 54 , row2 column 1 , negative 2 32 minus 166 , row3 column 1 , 1 minus 16 83 minus 220 bar , end table

  The quotient is x 2 − 16 x + 83 , R − 220 . x squared , minus 16 x plus 83 comma cap r minus 220 .

✓ Got It?

3. Use synthetic division to divide x 3 − 57 x + 56 x cubed , minus 57 x plus 56  by x − 7 . x minus 7 .  What is the quotient and remainder?

Problem 4 Using Synthetic Division to Solve a Problem

Crafts The polynomial x 3 + 7 x 2 − 38 x − 240 x cubed , plus 7 , x squared , minus 38 x minus 240  expresses the volume, in cubic inches, of the shadow box shown.

1. What are the dimensions of the box? (Hint: The length is greater than the height (or depth).)

   − 5 1 7 − 38 − 240 − 5 − 10 240 1 2 − 48 0 ¯ table with 3 rows and 1 column , row1 column 1 , negative 5 , 1 7 minus 38 minus 240 , row2 column 1 , negative 5 minus 10 , 240 , row3 column 1 , 1 2 minus 48 0 bar , end table

   x 2 + 2 x − 48 = ( x − 6 ) ( x + 8 ) x squared , plus 2 x minus 48 equals open x minus 6 close open x plus 8 close

   So , x 3 + 7 x 2 − 38 x − 240 = ( x + 5 ) ( x 2 + 2 x − 48 )   = ( x + 5 ) ( x − 6 ) ( x + 8 ) table with 2 rows and 2 columns , row1 column 1 , cap so comma , x cubed , plus 7 , x squared , minus 38 x minus 240 , column 2 equals . open , x plus 5 , close . open . x squared , plus 2 x minus 48 . close , row2 column 1 , , column 2 equals . open , x plus 5 , close . open , x minus 6 , close . open , x plus 8 , close , end table

   The length, width, and height (or depth) of the box are ( x + 8 ) open x plus 8 close  in., ( x + 5 ) open x plus 5 close  in., and ( x − 6 ) open x minus 6 close  in., respectively.