How to divide polynomials

1. 3.3 Dividing
Polynomials pg.
269

2. Division Algorithm pg. 269
If 𝑃(𝑥) and 𝐷(𝑥) are polynomials, with
𝐷(𝑥) ≠ 0, then there exists unique
polynomials 𝑄 𝑥 and 𝑅(𝑥), where 𝑅 𝑥 is
either 0 or of a degree less than the degree
of 𝐷(𝑥), such that
𝑃 𝑥 = 𝐷 𝑥 ∙ 𝑄 𝑥 + 𝑅(𝑥)
The polynomials 𝑃(𝑥) and 𝐷(𝑥) are called
the dividend and divisor, respectively, 𝑄(𝑥)
is the quotient, and 𝑅(𝑥) is the remainder

3. Synthetic Division pg. 271
Synthetic division is a quick method of
dividing polynomials; it can be used
when the divisor is of the form 𝑥 − 𝑐. In
synthetic division, we write only the
essential parts of the long division. Let’s
compare long and synthetic division on
the next slide

4. Synthetic Division pg. 271

5. Remainder Theorem pg. 272
If the polynomial 𝑃(𝑥) is divided by 𝑥 − 𝑐, then the
remainder is the value 𝑃(𝑐)
Proof
◦ If we divide 𝑃(𝑥) by 𝑥 − 𝑐, then the remainder must
be a constant since its degree must be less that 𝑥 −
𝑐. Let’s call the remainder 𝑟. We have
𝑃 𝑥 = 𝑥 − 𝑐 ∙ 𝑄 𝑥 + 𝑟
◦ If we replace 𝑥 with c, we have
𝑃 𝑐 = 𝑐 − 𝑐 ∙ 𝑄 𝑐 + 𝑟 = 0 + 𝑟 = 𝑟

6. Factor Theorem pg. 272
𝑐 is a zero of 𝑃 if and only if 𝑥 − 𝑐 is a factor of
𝑃(𝑥)
Proof
◦If 𝑃(𝑥) factors as 𝑃 𝑥 = 𝑥 − 𝑐 𝑄(𝑥), then
𝑃 𝑐 = 𝑐 − 𝑐 ∙ 𝑄 𝑥 = 0 ∙ 𝑄 𝑥 = 0
◦Conversely, if 𝑃 𝑐 = 0, then by the
Remainder Theorem
𝑃 𝑥 = 𝑥 − 𝑐 ∙ 𝑄 𝑥 + 0 = 𝑥 − 𝑐 ∙ 𝑄(𝑥)
◦So 𝑥 − 𝑐 is a factor of 𝑃(𝑥)