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5.4 synthetic division

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Part 2: Synthetic Division & The Remainder Theorem
Synthetic Division
Synthetic Division is a process that simplifies long
 division, but it can only be used when dividing a
 polynomial by a linear factor of the form x – a.
Synthetic Division
1.   Write the polynomial in standard form, including zero
     coefficients where appropriate
2.   Set up: use the opposite sign of a (this allows us to add
     throughout the process) and write the coefficients of
     the polynomial.
3.   Bring down the first coefficient
4.   Multiply the coefficient by the divisor. Add to the next
     coefficient.
5.   Continue multiplying and adding through the last
     coefficient.
6.   Write the quotient and remainder. The remainder will
     be the last sum.
Example: Divide using synthetic division
(x   3
         − 57 x + 56 ) ÷ ( x − 7 )
Example: Divide using synthetic division
(x   3
         − 14 x + 51x − 54 ) ÷ ( x + 2 )
               2
Example: Using synthetic division to solve a
problem
The polynomial x 3 + 7 x 2 − 38 x − 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)
2. If the width of the box is 15 in,
   what are the other dimension?

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5.4 synthetic division

  • 1. Part 2: Synthetic Division & The Remainder Theorem
  • 2. Synthetic Division Synthetic Division is a process that simplifies long division, but it can only be used when dividing a polynomial by a linear factor of the form x – a.
  • 3. Synthetic Division 1. Write the polynomial in standard form, including zero coefficients where appropriate 2. Set up: use the opposite sign of a (this allows us to add throughout the process) and write the coefficients of the polynomial. 3. Bring down the first coefficient 4. Multiply the coefficient by the divisor. Add to the next coefficient. 5. Continue multiplying and adding through the last coefficient. 6. Write the quotient and remainder. The remainder will be the last sum.
  • 4. Example: Divide using synthetic division (x 3 − 57 x + 56 ) ÷ ( x − 7 )
  • 5. Example: Divide using synthetic division (x 3 − 14 x + 51x − 54 ) ÷ ( x + 2 ) 2
  • 6. Example: Using synthetic division to solve a problem The polynomial x 3 + 7 x 2 − 38 x − 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) 2. If the width of the box is 15 in, what are the other dimension?
  • 7. The Remainder Theorem The Remainder Theorem provides a quick way to find the remainder of a polynomial long-division problem. If you divide a polynomial P(x) of degreen ≥ 1 by x − a then the remainder is P(a)
  • 8. Example: Evaluating a Polynomial Given that P ( x ) = x − 2x − x + 2 5 3 2 what is P(3)? By the remainder theorem, P(3) is the remainder when you divide P(x) by x – 3.
  • 9. Example: Evaluating a Polynomial Given that P ( x ) = x − 3 x − 28 x + 5 x + 20 5 4 3 what is P(─ 4)? By the remainder theorem, P(─ 4) is the remainder when you divide P(x) by x + 4.
  • 10. Homework P308 #21 – 39 odd, 40 – 43 all, 53 – 56 all, 57 – 61 odd