1. Section 5.4 – Dividing Polynomials

2. Long Division Recall division from arithmetic: Answer:
Since there is a remainder, we know that 3 is NOT a factor of 247.
Since there is a remainder of 0, we know that 2 IS a FACTOR of 48.

3. Long Division
1. For the function ℎ 𝑥 = 𝑥 3 − 𝑥 2 −17𝑥−15, use long division to determine which of the following are factors of ℎ(𝑥).
(a) 𝑥 (b) 𝑥+1
Both polynomials should be in standard form.
Since the remainder is NOT 0, 𝑥+5 is NOT a factor of
𝑥 3 − 𝑥 2 −17𝑥−15.

4. Long Division
Since the remainder is 0, 𝑥+1 is a factor of 𝑥 3 − 𝑥 2 −17𝑥−15.

5. Long Division Recall division from arithmetic: Answer: Check:
(divisor)(quotient) + (remainder) = dividend
(3)(82) + 1 = 247

6. Polynomial Division
2. A polynomial P(x) and a divisor d(x) are given. Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x), and express P(x) in the form P(x) = d(x) ∙Q(x) + R(x).
Both polynomials should be in standard form.

7. Synthetic Division
Use synthetic division to find the quotient and the remainder.
Both polynomials should be in standard form.

8. Synthetic Division
Use synthetic division to find the quotient and the remainder.
Both polynomials should be in standard form.

9. Synthetic Division
Use synthetic division to find the quotient and the remainder.
Both polynomials should be in standard form.

10. Remainder Theorem When 𝑃(𝑥) is divided by (𝑥 −𝑐), we write
So, the polynomial evaluated at 𝒙=𝒄 is equal to the remainder 𝑹(𝒄) when the polynomial is divided by (𝒙 −𝒄).

11. Remainder Theorem Use synthetic division to find the function value.
Polynomial should be in standard form.

12. Remainder Theorem Use synthetic division to find the function value.
Polynomial should be in standard form.

13. Remainder Theorem Use synthetic division to find the function value.
Polynomial should be in standard form.

14. Factor Theorem
For a polynomial 𝑃(𝑥), if 𝑃 𝑐 =0, then c is a zero of the polynomial function and (𝑥 −𝑐) is a factor.
Using synthetic division, determine whether the numbers are zeros of the polynomial function.
Polynomial should be in standard form.