Rational Zeros Theorem Upper & Lower Bounds Long Division Remainder Theorem Factor Theorem Synthetic Division 100 100 100 100 100 100 200 200 200 200 200 200 300 300 300 300 300 300 400 400 400 400 400 400 500 500 500 500 500 500

Division involves a dividend divided by a divisor to obtain a quotient and a _________. remainder

2x4-x3-x=(2x2+x+1)(x2-2)+(x-2) What is the answer of 2x4-x3-2 divided by 2x2+x+1? Write in polynomial form. 2x4-x3-x=(2x2+x+1)(x2-2)+(x-2)

Divide f(x)=x2-2x+3 b x-1. Write in polynomial form.

Divide f(x)=x3+4x2+7x-9 by x+3 and write a summary statement in polynomial form.

f(x)=(x2-4x+12)(x2+2x-1)-32x+18 Divide f(x)=x4-2x3+3x2-4x+6 by x2+2x-, write a summary statement in fraction form. f(x)=(x2-4x+12)(x2+2x-1)-32x+18

If a polynomial is divided by ___, then the remainder is r=f(k). x-k

r=f(-1)=3(-1)2+7(-1)-20=3-7-20=-24 Using the remainder theorem, find the remainder when f(x)=3x2+7x-20 is divided by x+1. r=f(-1)=3(-1)2+7(-1)-20=3-7-20=-24

What is the remainder when f(x)=2x2-3x+1 is divided by x-2?

Use the remainder theorem to find the remainder when you divide f(x)=x3-x2+2x-1 by x+3 -43

Use the remainder theorem to find the remainder when f(x)=2x3-3x2+4x-7 by x-2 5

A polynomial function f(x) has a factor x-k if and only if f(k)=_.

Using the factor theorem, factor f(x)=3x2+7x-20 by dividing it by the known factor x+4.

Is x-1 a factor of x3-x2+x-1? Yes

Use the factor theorem to determine whether x-2 is a factor of x3+3x-4 No

Use the factor theorem to determine where x+2 is a factor of 4x3+9x2-3x-10 Yes

This shortcut method for the division of a polynomial by a ______ _______ x-k is synthetic division. linear divisor

Divide 2x3-3x2-5x-12 by x-3 using synthetic division and write a summary statement in fraction form. 2x3-3x2-5x-12 = 2x2+3x+4 x-3

Divide using synthetic division and write a summary statement in fraction form: x3-5x2+3x-2 x+1

Divide using synthetic division and write a summary statement in fraction form: 9x3+7x2-3x x-10

Divide using synthetic division and write a summary statement in fraction form: 5x4-3x+1 4-x

Real zeros of a polynomial function are either ________ zeros – zeroes that are rational numbers – or __________ zeros – zeros that are irrational numbers. rational, irrational

Find the rational zeros of f(x)=x3-3x2+1 So f has no rational zeros.

(all positive & negative) Use the rational zeroes theorem to write a list of all possible rational zeroes of f(x)=6x3-5x-1 1 1, 2, 3, 6 (all positive & negative)

(all positive & negative) Use the rational zeroes theorem to write a list of possible rational zeroes: f(x)=2x3-x2-9x+9 1, 3, 9 1, 2 (all positive & negative)

Find all the zeroes of the function and identify each zero as rational or irrational: f(x)=2x4-7x3-2x2-7x-4 -1/2 and 4; rational No irrational

We narrow our search for real zeros by using a test that identifies upper and lower ______ for real zeros. bounds

Find a lower bound of f(x)=2x4-7x3-8x2+14x+8 -2

Use synthetic division to prove that to number k is an upper bound for the real zeroes of the function: k=3; f(x)=2x3-4x2+x+3 Yes

Use synthetic division to prove that the number k is an upper bound for the real zeroes of the function f: k=2; f(x)=x4-x3+x2+x-12 Yes

Use synthetic division to prove that the number k is a lower bound for the real zeroes of the function f: k=-1; f(x)=3x2-4x2+x-2 Yes