Section 5.4 – Dividing Polynomials

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Section 5.4 – Dividing Polynomials
Name: ________________________ Date: ___________________ Period: ____________
Algebra 2 Section 5.4 Dividing Polynomials Notes
Part 1: Long Division We can divide polynomials using traditional long division. (Hint: Dead Mice Smell Bad)
Example 1:
(2𝑥 3 + 11𝑥 2 + 9𝑥 + 2) ÷ (2𝑥 + 1)
Example 2:
(𝑥 3 + 3𝑥 2 − 4𝑥 − 12) ÷ (𝑥 − 2)
Example 3:
5𝑥 4 +2𝑥 3 −9𝑥+12
𝑥 2 −𝑥+4
***Use placeholders for all missing terms.***
Example 4:
𝑥 3 +𝑥−10
𝑥+3
***We write remainders as mixed numbers.***
Part 2: Synthetic Division- A short cut
When dividing x-k, where k is any real number, you can use synthetic division.
Example 5:
(5𝑥 3 + 6𝑥 + 8) ÷ (𝑥 + 2)
Example 6:
(4𝑥 4 − 2𝑥 2 + 𝑥 + 1) ÷ (𝑥 − 1)
The Remainder Theorem: If a polynomial f(x) is divided by (x – k), then the remain is r = f(k).
 If k is the input value, then the remainder is the output value.
Example 7: Find f(x) given that 𝑓(𝑥) = 3𝑥 3 + 8𝑥 2 + 5𝑥 − 7 and x= – 2.
Example 8: 𝑔(𝑥) = 4𝑥 3 + 10𝑥 2 − 3𝑥 − 8; 𝑔(−1)
The Factor Theorem: A polynomial f(x) has a factor (x – k) if and only if f(k)=0.
 If the remainder is zero, then (x – k) is a factor
Example 9: Show that (x – 2) and (x – 3) are factors of 𝑓(𝑥) = 2𝑥 4 + 7𝑥 3 − 4𝑥 2 − 27𝑥 − 18. Then find the
remaining factors and list all zeroes.
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Example 10: Given that 𝑥 = 2 is a zero of the function 𝑓(𝑥) = 2𝑥 3 − 15𝑥 2 + 34𝑥 − 21 find all of the factors
and list all the zeroes.

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