Please Check your HW- Period 7

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Presentation transcript:

Please Check your HW- Period 7

Please Check your HW- Period 7

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.

Polynomial Long Division 1) Factor 6x 3 + 17x 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

Polynomial Long Division 2) Factor 6x 3 + x 2 – 117x + 140 completely using long division if (3x – 4) is a factor. (3x – 4)(x + 5)(2x – 7)

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).

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

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

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

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

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

Synthetic Division 2) Find (8x 4 + 38x 3 + 5x 2 + 3x + 3) ÷ (4x + 1) using synthetic division.

Synthetic Division 3) Find (6x 4 + 11x 3 - 15x 2 – 12x + 7) ÷ (3x + 1) using synthetic division.

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 2 + 1000x + 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 –150 1000 480,000 –750 1250 – 150 250 481,250 $481,250

The Factor Theorem 1) Use the Factor Theorem to determine if (x – 5) and (x + 5) are factors of f (x) = x 3 – 18x 2 + 60x + 25. Use the binomials that are factors to write a factored form of f (x). –5 1 –18 60 25 –5 115 –875 1 –23 175 –850 f (x) = (x – 5)(x 2 – 13x – 5)

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)

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 + 30. Use the binomials that are factors to write a factored form of f (x). yes, yes; f(x) = (x + 2)(x – 3)(4x – 5)

2 More for Practice!