Synthetic Substitution Worksheet



Synthetic Division Reminder!

A few examples of how it is done…

(1x4 – 4x3 – 7x2 + 34x – 24) ÷ (x + 3)

-3 1 -4 -7 34 -24

-3 21 -42 24

1 -7 14 -8

So, the answer would be 1x3 – 7x2 + 14x – 8

(2x5 – 14x3 + 24x) ÷ (x – 3)

So, let’s rewrite it with those in there.

(2x5 + 0x4 – 14x3 + 0x2 + 24x + 0) ÷ (x – 3)

Now we can do the work.

3 2 0 -14 0 24 0

6 18 12 36 180

2 6 4 12 60 180

So, the answer would be 2x4 + 6x3 + 4x2 + 12x + 60 +

Polynomial Division Reminder!

A few examples of how it is done…

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Name______________________________________________________________Date_____________

Divide the following using both methods of Polynomial Division and Synthetic Substitution:

|Polynomial Division |Synthetic Substitution |

|(x3 – 7x – 6) ÷ (x – 2) |(x3 – 7x – 6) ÷ (x – 2) |

|(4x2 + 5x + 8) ÷ (x + 1) |(4x2 + 5x + 8) ÷ (x + 1) |

|(x3 – 14x + 8) ÷ (x + 4) |(x3 – 14x + 8) ÷ (x + 4) |

|(x2 + 10) ÷ (x + 4) |(x2 + 10) ÷ (x + 4) |

Divide the following using the method that you prefer! (Polynomial Division or Synthetic Substitution)

|(10x4 + 5x3 + 4x2 – 9) ÷ (x + 1) |(x3 + 8x2 – 3x + 16) ÷ (x + 5) |

|(2x4 – 6x3 + x2 – 3x – 3) ÷ (x – 3) |(x4 – 6x3 – 40x + 33) ÷ (x – 7) |

|(4x4 + 5x3 + 2x2 – 1) ÷ (x + 1) |(-10x5 + 3x – 7) ÷ (x – 1) |

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EX. 1

You set it up like this…

Whatever number ends up in this position is the remainder. With this case, x + 3 divided into the polynomial evenly.

x + 3 = 0

-3 -3

x = -3

Once you finish you put the variable x back onto the number starting one degree lower that it was before.

In this case you start with x3 because it was x4 before the synthetic division.

0

Anytime you are missing exponents you MUST add in the missing terms. So, if the highest exponent is x5, then you should see an x4, x3, x2, x, and a constant term.

Anytime you are missing exponents you MUST add in the missing terms. So, if the highest exponent is x5, then you should see an x4, x3, x2, x, and a constant term.

Anytime you are missing exponents you MUST add in the missing terms. So, if the highest exponent is x5, then you should see an x4, x3, x2, x, and a constant term.

EX. 2

When you have a remainder, you add the remainder onto the answer and put it over what they were dividing by.

With the remainder, you add it onto the answer and put it over what they were dividing by.

With the remainder, you add it onto the answer and put it over what they were dividing by.

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