1. Polynomial Long Division
Algebraically finding the roots and factors of polynomials.

2. Remember…

3. Polynomials 𝑝(𝑥 𝑑(𝑥 =𝑞(𝑥)+ 𝑟(𝑥) d(𝑥) 𝑝(𝑥)=𝑞(𝑥)𝑑(𝑥)+𝑟(𝑥
We can do the same thing with polynomials. Let our first polynomial be p(x), our divisor (the one we are dividing by) be d(x), or quotient (answer) is q(x) and our remainder is r(x) then… ALSO
𝑝(𝑥 𝑑(𝑥 =𝑞(𝑥)+ 𝑟(𝑥) d(𝑥)
𝑝(𝑥)=𝑞(𝑥)𝑑(𝑥)+𝑟(𝑥

4. Example #1 - Find the quotient q(x) and the remainder r(x) if p(x) is divided by d(x). Write your answer in the form p(x)=q(x)d(x)+r(x):
𝑝(𝑥)= 𝑥 2 −11𝑥+30
𝑑(𝑥)=𝑥−3

5. Example #1 - Find the quotient q(x) and the remainder r(x) if p(x) is divided by d(x). Write your answer in the form p(x)=q(x)d(x)+r(x):
𝑝(𝑥)= 𝑥 2 −11𝑥+30
𝑑(𝑥)=𝑥−3 x -8 -3 x2
-11x
+30
-x2
+3x ↓
-8x 8x
-24 6

6. Note about Remainders
When dividing, if the remainder is zero, that means that our divisor is a FACTOR of our dividend. Polynomial long division can be used to factor out complex polynomials.

7. Rational Roots Theorem
Let f(x) be a polynomial. Then p/q is a potential zero of f(x) if: q is a factor of the leading coefficient and p is a factor of the constant term. p = Factors of the last term q = Factors of the first term

8. Example #2 – List the potential rational zeros of f(x).

9. Example #2 – List the potential rational zeros of f(x).

10. Assignment
Polynomial Long Division Worksheet