Opening Quiz:

• Sketch a graph of the following polynomial function by hand using the methods discussed on Friday (make sure to label and be specific as possible) :

Long Division of Polynomials

• If you're dividing a polynomial by something more complicated than just a simple monomial, then you'll need to use a different method for the simplification. That method is called "long (polynomial) division", and it works just like the long (numerical) division you did back in elementary school, except that now you're dividing with variables.

Long Division of Polynomials

• Divide by

Student Check:

• Answer the following:

Try Again:

• Answer:

Class Opener:

• Solve the following problems using long division:

Synthetic Division

• There is a nice shortcut for long division of polynomials when dividing by divisors of the for (x – k). This short cut is known as SYNTHETIC DIVISION

Let’s look at how to do this using the example:

4 25 4 6 ( 3)x x x x In order to use synthetic division these

two things must happen:There must be a coefficient for every possible power of the

variable.

The divisor must have a leading coefficient of 1.

#1 #2

Step #1: Write the terms of the polynomial so the degrees are in descending order.

4 3 25 0 4 6x x x x

3

Since the numerator does not contain all the powers of x,

you must include a for the .0 x

Step #2: Write the constant a of the divisor x- a to the left and write down the

coefficients.

Since the divisor , then 3 3 x a

4 3 25 0 4 6

3 5 0 4 1 6

x x x x

Step #3: Bring down the first coefficient, 5.

3 5 0 4 1 6

5

Step #4: Multiply the first coefficient by r (3*5). 3 5 0 4 1 6

15

5

Step #5: After multiplying in the diagonals, add the column.

3 5 0 4 1 6

15

5 15

Add the column

Step #6: Multiply the sum, 15, by ; 15 3=15,

and place this number under the next coefficient,

then add the column again.

r

3 5 0 4 1 6

15 45

5 15 41

Multiply the diagonals, add the columns.

Add

41

Step #7: Repeat the same procedure as step #6.

3 5 0 4 1 6

15 45 123 372

5 15 41 12 784 3

Add Columns

Add Columns

Add Columns

Add Columns

Step #8: Write the quotient.

The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

The quotient is:

5x3 15x2 41x 124 378

x 3

Remember to place the remainder over the divisor.

Try this one:

3 21) ( 6 1) ( 2)t t t

2 311 8 16

2Quo i tt t

tent

2 1 6 0 1

2 16 32

1 8 16 31

Student Check

Use synthetic division to divide

Exit Slip:

• Use synthetic division to divide:

Remainder Theorem

• If a polynomial f(x) is divided by x – k, the remainder is r = f(k)

Using Remainder Theorem:

• Use the Remainder Theorem to evaluate the following function at x = -2

Factor Theorem:

• A polynomial f(x) has a factor (x-k) if and only if f(k) = 0

Factoring a Polynomial: Repeat Division

• Show that (x – 2) and (x+3) are factors of

Then find the remaining factors of f(x)

Student Check:

• Show that (x -5) and (x+4) are factors of

Then find the remaining factors of f(x)

Using the Remainder in Synthetic Division:

• In summary, the remainder r, obtained in the synthetic division of f(x) by (x – k) provides the following information:

1. If r = 0, (x – k) is a factor of f(x)2. The remainder, r, gives the value of f at x=k

that is, r = f(k).3. If r = 0, (k,0) is an x-intercept of the graph of f

Rational Zero Test:

• The rational zero test relates the possible rational zeros of a polynomial (having integer coefficients) to the leading coefficient and to the constant term of the polynomial.

Using the Rational Zero Test:

• To us the rational zero test, first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient.

Tips:

• If the leading coefficient is not 1, the list of possible zeros can increase dramatically. Use these tips to shrink your search:

1. Use a graphing calculator to graph the polynomial

2. Use IVT, and table function of calculator3. Use The Factor Theorem and synthetic

division.

Using the Rational Zeros Test:

• Find the rational zeros of

Student Check:

• Find the rational zeros of the following:

Descartes’ Rule of Signs

• method of determining the maximum number of zeros in a polynomial.

Look on page 124 in text.