Warm - Up Divide using long division 150 / 7 7

Announcements Assignment p. 239 # 1, 7, 8 - 20 even 2.1 – 2.3 Quiz Friday Please pass up last night’s homework

2.3 Polynomial and Synthetic Division Kenwood Students will be able to: use long division to divide polynomials by other polynomials use synthetic division to divide polynomials by binomials of the form (x  k)

Why spend our time on it? If we are given one x-intercept, we can find the others using division

Long Division to divide polynomials by other polynomials

Procedure Divide first terms Multiply times the divisor Subtract by changing the signs and adding Bring down the next term and begin the process again

1.Divide first terms Try to make the first terms equivalent so they will cancel 2x

2.Multiply times the divisor 2x

3. Subtract by changing the signs and adding 2x 4x

4. Bring down the next term and begin the process again

Result??? =2x + 4 2x +4

Example 1. Divide 2x3 – 5x2 + x – 8 by x – 3. Quotient Dividend Divisor Remainder

Division Algorithm How do we write our answer?  f(x) = function  d(x) =Divisor  q(x) = quotient  r(x) = remainder Please go back to the first example and label f(x), d(x), q(x) and r(x) in the first problem

Example 2. Please find the values of a, b, c

Example 3. Divide 5x2 – 17x - 12 by x – 4. Please complete this question on a separate sheet of paper to be turned in.

Warm – Up from the assignment 16) 18) 20)

Announcements 2.1 – 2.3 Quiz Friday Assignment p. 239 # 23, 24 – 32 even, 49 2.1 – 2.3 Quiz Friday

Synthetic Division

Procedure Find “a”, what x equals Write the coefficients and “a” in the synthetic division format Bring down the first coefficient. Then multiply and add for each column. Write the quotient

Example # 1

Step 1 Find “a”, what x equals (x – 2) = 0 +2 +2 X = 2

Step 2 Write the coefficients and “a” in the synthetic division format Drop down the coefficients into a line 6 -19 16 -4 2

Step 3 Bring done the first coefficient. Then multiply and add for each column. 6 -19 16 -4 12 -14 4 6 7 2 0 2

Step 4 Write the quotient 6 -19 16 -4 12 -14 4 6 7 2 0 2

Comparing Long Division and Synthetic Division Purpose Use when you have a division problem involving two polynomials. Used only in the case of division by a linear factor Advantage Will work for any division problem Shorter process Disadvantage Longer work process Longer Set - Up Set - Up Set up exactly like an integer division problem Only the coefficients of the polynomial are used

Example 2. Divide 3x3 – x2 + 5x – 3 by x – 2. How is Synthetic Division connected to Long Division? -2 3 -1 2 -3 ( - ) -6 -10 -24 5 12 21

Example #3 What to we have to do to keep the place holders?

Example # 2

Procedure – Step 1 Find “a”, what x equals (x – 3) = 0 +3 +3 X = 3

Step 2 Write the coefficients and “a” in the synthetic division format 2 -3 -10 7 3

Step 3 Bring done the first coefficient. Then multiply and add for each column. 2 -3 -10 7 6 9 -3 2 3 -1 4 3

Step 4 Write the quotient 2 -3 -10 7 6 9 -3 2 3 -1 4 3

Warm - Up 1) 2)

Announcements Assignment 2.1 – 2.3 Quiz tomorrow p. 239 # 11, 17, 23, 27, 45, 47 Review notes from 2.1 and 2.2 2.1 – 2.3 Quiz tomorrow

Factor Theorem If a number (k) is a factor, then there will be no remainder. How could we test for that? Substitute the value into the function and find out if there is a remainder Are 1, 2, 0, -1 factors of f(x) = x4 – 1?

Remainder Theorem The remainder for any division problem for a number (k) will be f(k) What is the remainder if x = 1?

Things to know for the 2.1 – 2.3 Quiz How to use long division to solve How to use synthetic division to solve Find the equation of a quadratic function given the vertex and a point Find the vertex given an equation Find the real zeros of a polynomial How to use the factor theorem How to use the remainder theorem

Cooperative Assignment Please sit quietly in groups of four (4). On your own sheet of paper, work on one of the 4 questions alone for 3 minutes. When time is up, pass your question slips to your left to the next person in the group. Continue step 3 until all 4 group members have completed each question. Next, you will share your answers with the other students in their groups, one question at a time. The group should come to an agreement about the correct answer to the question. When you have agreed on an answer to a question, put the work on a post-it with the following info: question #, group name, agreed-upon answer to the question. Do this for all four questions. Have one person from your group put the 4 questions on the board.