Long Division AlgorithmLong Division Algorithmand Synthetic Division!!!and Synthetic Division!!!

Sec. 3.3aSec. 3.3a

Homework: p. 373-374 1-31 oddHomework: p. 373-374 1-31 odd

First, let’s work through this…First, let’s work through this…

32 35871

3832

1

326764

2

3Remainder!

We can write our answer as:

32 112 3 3587

We use a similar processWe use a similar processwhen dividing when dividing polynomialspolynomials!!!!!!

Division Algorithm for PolynomialsDivision Algorithm for PolynomialsLet f(x) and d(x) be polynomials with the degree of f greaterthan or equal to the degree of d, and d(x) = 0. Then there areunique polynomials q(x) and r(x), called the quotient andremainder, such that

f x d x q x r x where either r(x) = 0 or the degree of r is less than the degreeof d. The function f(x) is the dividend, d(x) is the divisor, andif r(x) = 0, we say d(x) divides evenly into f(x).

Fraction form:

f x r x

q xd x d x

Using Polynomial Long DivisionUsing Polynomial Long DivisionUse long division to find the quotient and remainder when is divided by . Write a summarystatement in both polynomial and fraction form.

4 32 2x x 22 1x x

2 4 3 22 1 2 0 0 2x x x x x x

2x

4 3 22x x x 3 22 0 2x x x

x

3 22x x x 2x

Quotient

Remainder

Using Polynomial Long DivisionUsing Polynomial Long DivisionUse long division to find the quotient and remainder when is divided by . Write a summarystatement in both polynomial and fraction form.

4 32 2x x 22 1x x

Can we verify these answers Can we verify these answers graphicallygraphically??????

Polynomial Form:

4 3 2 22 2 2 1 2x x x x x x x Fraction Form:

4 32

2 2

2 2 2

2 1 2 1

x x xx x

x x x x

Synthetic DivisionSynthetic Division

Synthetic DivisionSynthetic Division is a shortcut method for the is a shortcut method for thedivision of a polynomial by a linear divisor, division of a polynomial by a linear divisor, xx – – kk..

Notes:Notes:

This technique works This technique works only only when dividing by awhen dividing by alinear polynomial…linear polynomial…

It is essentially a “collapsed” version of the longIt is essentially a “collapsed” version of the longdivision we practiced last class…division we practiced last class…

Synthetic Division – Examples:Synthetic Division – Examples:3 22 3 5 12x x x

3x Evaluate the quotient:Evaluate the quotient:

33 22 ––33 ––55 ––1212

22 33 44 00

66 99 1212

Coefficients of dividend:Coefficients of dividend:Zero ofZero ofdivisor:divisor:

RemaindeRemainderr

QuotientQuotient3 22 3 5 12x x x

3x 22 3 4x x

Synthetic Division – Examples:Synthetic Division – Examples:4 22 3 3x x x 2x Divide by and write a

summary statement in fraction form.

––22 11 00 ––22 33

4 23 22 3 3 1

2 2 12 2

x x xx x x

x x

––33

11 ––22 22 ––11 ––11

––22 44 ––44 22

Verify Graphically?Verify Graphically?

PracticePractice

23 7 20f x x x

Divide the above function by 4x

Divide the above function by 3 5x

Using Our New TheoremsUsing Our New Theorems

Is the first polynomial and factor of the second?

3x 3 2 15x x x

Yes, Yes, xx – 3 is a factor of the second – 3 is a factor of the secondpolynomialpolynomial

Some whiteboard problems…

4 3 2 2

Divide and write a summary statement in polynomial form and fraction form:

( ) 3 6 3 5 ( ) 1f x x x x x d x x

4 3 2

Divide and write a summary in fraction form:

2 5 7 3 1

3

x x x x

x

““Fundamental Connections” forFundamental Connections” for Polynomial FunctionsPolynomial Functions

For a polynomial function f and a real number k, the followingstatements are equivalent:

1. x = k is a solution (or root) of the equation f(x) = 0.

2. k is a zero of the function f.

3. k is an x-intercept of the graph of y = f(x).

4. x – k is a factor of f(x).