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Worksheet: Evaluating Polynomial Functions Using Synthetic Substitution

Q1:

Consider the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 9 π‘₯ + 3 π‘₯ βˆ’ 7 π‘₯ + 1 2 4 3 2 .

Use synthetic division to find the quotient 𝑄 ( π‘₯ ) and the remainder 𝑅 satisfying 𝑓 ( π‘₯ ) = 𝑄 ( π‘₯ ) ( π‘₯ + 2 ) + 𝑅 .

  • A 𝑄 ( π‘₯ ) = π‘₯ βˆ’ 1 1 π‘₯ + 2 5 π‘₯ βˆ’ 5 7 3 2 , 𝑅 = 1 0 2
  • B 𝑄 ( π‘₯ ) = π‘₯ βˆ’ 7 π‘₯ βˆ’ 1 1 π‘₯ βˆ’ 2 9 3 2 , 𝑅 = βˆ’ 4 6
  • C 𝑄 ( π‘₯ ) = π‘₯ βˆ’ 7 π‘₯ βˆ’ 1 1 π‘₯ βˆ’ 2 9 3 2 , 𝑅 = βˆ’ 7 0
  • D 𝑄 ( π‘₯ ) = π‘₯ βˆ’ 1 1 π‘₯ + 2 5 π‘₯ βˆ’ 5 7 3 2 , 𝑅 = 1 2 6
  • E 𝑄 ( π‘₯ ) = π‘₯ + 7 π‘₯ βˆ’ 1 1 π‘₯ + 1 5 3 2 , 𝑅 = βˆ’ 4 2

Find 𝑓 ( βˆ’ 2 ) .

Q2:

Use synthetic substitution to find the value of 𝑓 ( 2 0 ) given 𝑓 ( π‘₯ ) = 0 . 0 6 π‘₯ βˆ’ 0 . 1 4 π‘₯ βˆ’ 3 . 1 π‘₯ + 5 . 4 4 3 .

Q3:

Consider the function 𝑓 ( π‘₯ ) = π‘₯ + 5 π‘₯ βˆ’ 8 π‘₯ + 9 4 3 2 .

What does the remainder theorem tell us about 𝑓 ( 3 ) ?

  • A 𝑓 ( 3 ) is the remainder when we divide 𝑓 ( π‘₯ ) by 3 π‘₯ βˆ’ 3 .
  • B 𝑓 ( 3 ) is the remainder when we divide 𝑓 ( π‘₯ ) by π‘₯ + 3 .
  • C 𝑓 ( 3 ) is the remainder when we divide 𝑓 ( π‘₯ ) by π‘₯ .
  • D 𝑓 ( 3 ) is the remainder when we divide 𝑓 ( π‘₯ ) by π‘₯ βˆ’ 3 .

Hence, use synthetic division to find 𝑓 ( 3 ) .