1. Please Check your HW- Period 7

2. Please Check your HW- Period 7

3. 2.3 Remainder and Factor Theorems
You factored quadratic or “quadratic-like” expressions to solve.
Divide polynomials using long division and synthetic division.
Use the Remainder and Factor Theorems.

4. Polynomial Long Division
1) Factor 6x x 2 – 104x + 60 completely using long division if (2x – 5) is a factor.
(–)6x 3 – 15x 2
(2x – 5)(3x – 2)(x + 6)
32x 2 – 104x
(–)32x 2 – 80x
–24x + 60
(–)–24x + 60

5. Polynomial Long Division
2) Factor 6x 3 + x 2 – 117x completely using long division if (3x – 4) is a factor.
(3x – 4)(x + 5)(2x – 7)

6. The Remainder Theorem
Let f(x) and d(x) be polynomials such that the degree of d(x) is less than or equal to f(x) and d(x)≠0.
Then, there exists a quotient q(x) and sometimes a remainder r(x) such that
When the remainder is 0, then d(x) divides evenly into f(x).

7. Non-Zero Remainder 3) Divide 6x 3 – 5x 2 + 9x + 6 by 2x – 1.
Be careful here!

8. Division of Degree 2 or Higher
5) Divide x 3 – x 2 – 14x + 4 by x 2 – 5x + 6.
(–)x 3 – 5x 2 + 6x
4x 2 – 20x + 4
(–)4x 2 – 20x + 24
–20

9. Division of Degree 2 or Higher
6) Divide 2x 4 + 9x 3 + x2 – x + 26 by x 2 + 6x + 9.

10. Synthetic Division
What happens if we are missing a term in the polynomial?

11. Synthetic Division
1) Find (2x 5 – 4x 4 – 3x 3 – 6x 2 – 5x – 8) ÷ (x – 3) using synthetic division.

12. Synthetic Division
2) Find (8x x 3 + 5x 2 + 3x + 3) ÷ (4x + 1) using synthetic division.

13. Synthetic Division
3) Find (6x x x 2 – 12x + 7) ÷ (3x + 1) using synthetic division.

14. DO NOW: Synthetic Division
Suppose 800 units of beachfront property have tenants paying $600 per week. For each $10 decrease in rent, 15 more units would be rented. The weekly revenue from the rentals is R (x) = –150x x + 480,000, where x is the number of $10 decreases the property manager is willing to take.
Use the Remainder Theorem to find the revenue from the properties if the property manager decreases the rent by $50.
5 – ,000 –
– ,250
$481,250

15. The Factor Theorem
1) Use the Factor Theorem to determine if (x – 5) and (x + 5) are factors of f (x) = x 3 – 18x x Use the binomials that are factors to write a factored form of f (x).
–5 1 –
–5 115 –875
1 – –850
f (x) = (x – 5)(x 2 – 13x – 5)

16. The Factor Theorem
2) Use the Factor Theorem to determine if (x – 5) and (x + 2) are factors of f (x) = x 3 – 2x 2 – 13x – 10. Use the binomials that are factors to write a factored form of f (x).
f (x) = (x – 5)(x + 2)(x + 1)

17. The Factor Theorem
3) Use the Factor Theorem to determine if the binomials (x + 2) and (x – 3) are factors of f (x) = 4x 3 – 9x 2 – 19x Use the binomials that are factors to write a factored form of f (x).
yes, yes; f(x) = (x + 2)(x – 3)(4x – 5)

18. 2 More for Practice!