Opening Questions 6 5 The Remainder and Factor

Opening Questions •

6. 5: The Remainder and Factor Theorems Objective: Divide polynomials and relate the result to the remainder theorem and the factor theorem Use polynomial division in real life problems

Polynomial Long Division *Multiply by the divisor • 2 x 2 + 7 x + 10 – ( ) 2 x 4 – 4 x 3 + 4 x 2 *Take the first term of the divisor and see what we would have to multiply by to match the first term of the numerator *Subtract *Repeat 7 x 3 – 4 x 2 + 5 x – ( ) 7 x 3 – 14 x 2 + 14 x 10 x 2 – 9 x – 1 – ( ) 10 x 2 – 20 x + 20 Remainder, put this over original divisor 11 x – 21

Ex 2 *Multiply by the divisor y 2 – y + 2 – ( ) – y 3 + y 2 + y – y 3 – y 2 – y – ( ) 2 y 2 + 2 y +5 – ( ) 2 y 2 + 2 y + 2 Remainder, put this over original divisor 3

Now Try •

Using Synthetic Division • *Use the opposite of the constant term 2 1 1 *Rewrite the equation. All your answers from the synthetic division are the coefficients. Your exponent is now one less 2 -6 -9 2 8 4 4 2 -5 Remainder

Ex 2 • 1 3 1 0 -10 -24 3 9 -3 3 -1 -27

Ex 3 -3 1 1 2 -6 -9 -3 3 9 -1 -3 0 *Remainder is zero. This means x+3 is a factor of x 3 + 2 x 2 – 6 x - 9 2 – x – 3. Therefore, we can say = x

Now Try •

Given a Zero of a Polynomial Ex 1 Factor f(x) = 2 x 3 + 11 x 2 + 18 x + 9 given f(-3) = 0 *Because f(-3) = 0, that means when x = -3, the function will equal zero. Which means x+3 is a factor -3 2 2 (x + 3)(2 x 2 + 5 x + 3) *Can factor more 11 18 9 -6 -15 -9 5 3 0 (x + 3)(2 x + 3)(x + 1)

Ex 2 Find all the zeros of the function f(x) = x 3 – 2 x 2 – 9 x + 18 if x = 2 is one zero 2 1 1 -2 -9 18 2 0 -18 0 -9 0 f(x) = (x – 2)(x 2 – 9) f(x) = (x – 2)(x + 3)(x – 2)

Now Try Find the factors given the zero 1) x 3 + x 2 – 10 x + 8 given f(-4) = 0 2) f(x) = 3 x 3 + 14 x 2 – 28 x -24 given f(-6) = 0 Find all the zeros of the function 3) f(x) = x 3 + 3 x 2 – 34 x + 48 given f(3) = 0 4) f(x) = 2 x 3 – 9 x 2 – 32 x – 21 given f(7) = 0

Real World Application You are an accountant for a manufacturer of radios. The demand function for the radios is p = 40 – 4 x 2 where x is the number of radios produced in millions. It costs the company $15 to make a radio. a) Write an equation giving profit as a function of the number of radios produced b) The company currently produces 1. 5 Million radios and makes a profit of $24 million, but you would like to scale back production. What lesser number of radios could the company produce to yield the same profit

Real World Application Continued a) Profit = Revenue – Cost P = (40 – 4 x 2)(x) – (15)(x) P = 40 x – 4 x 3 – 15 x P = -4 x 3 + 25 x

Real World Application Continued b) P = -4 x 3 + 25 x 24 = -4 x 3 + 25 x 0 = -4 x 3 + 25 x - 24 *We know 1. 5 million is already one answer, or one zero 1. 5 -4 -4 0 25 24 -6 -9 -24 16 0 -6 *Rewrite as factors (x – 1. 5)(-4 x 2 – 6 x + 16) *Simplify (x – 1. 5) -2(2 x 2 + 3 x – 8) Can’t have negative units, so the company must make about 1. 39 million radios to produce the same profit

Now Try A company that manufactures CD-ROM drives would like to increase its production. The demand function for the drivers is p = 75 – 3 x 2, where p is the price the company charges per unit when the company produces x million units. It costs the company $25 to produce each drive a) Write an equation giving the company’s profit as a function of the number of CD-ROM drives it manufactures b) The company currently manufactures 2 million CD-ROM drives and makes a profit of $76 million. At what other level of production would the company also make $76 million?