Worksheet: Evaluating Polynomial Functions Using Synthetic Substitution

In this worksheet, we will practice using synthetic substitution to evaluate polynomial functions.

Q1:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 + 5 𝑥 8 𝑥 + 9 4 3 2 .

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

  • A 𝑓 ( 3 ) is the remainder when we divide 𝑓 ( 𝑥 ) by 3 𝑥 3 .
  • B 𝑓 ( 3 ) is the remainder when we divide 𝑓 ( 𝑥 ) by 𝑥 + 3 .
  • C 𝑓 ( 3 ) is the remainder when we divide 𝑓 ( 𝑥 ) by 𝑥 .
  • D 𝑓 ( 3 ) is the remainder when we divide 𝑓 ( 𝑥 ) by 𝑥 3 .

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

Q2:

Use synthetic substitution to find the value of 𝑓 ( 2 0 ) given 𝑓 ( 𝑥 ) = 0 . 0 6 𝑥 0 . 1 4 𝑥 3 . 1 𝑥 + 5 . 4 .

Q3:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 9 𝑥 + 3 𝑥 7 𝑥 + 1 2 4 3 2 .

Use synthetic division to find the quotient 𝑄 ( 𝑥 ) and the remainder 𝑅 satisfying 𝑓 ( 𝑥 ) = 𝑄 ( 𝑥 ) ( 𝑥 + 2 ) + 𝑅 .

  • A 𝑄 ( 𝑥 ) = 𝑥 1 1 𝑥 + 2 5 𝑥 5 7 3 2 , 𝑅 = 1 0 2
  • B 𝑄 ( 𝑥 ) = 𝑥 7 𝑥 1 1 𝑥 2 9 3 2 , 𝑅 = 4 6
  • C 𝑄 ( 𝑥 ) = 𝑥 7 𝑥 1 1 𝑥 2 9 3 2 , 𝑅 = 7 0
  • D 𝑄 ( 𝑥 ) = 𝑥 1 1 𝑥 + 2 5 𝑥 5 7 3 2 , 𝑅 = 1 2 6
  • E 𝑄 ( 𝑥 ) = 𝑥 + 7 𝑥 1 1 𝑥 + 1 5 3 2 , 𝑅 = 4 2

Find 𝑓 ( 2 ) .

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