Long Division Algorithm and Synthetic Division!!!

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Presentation transcript:

Long Division Algorithm and Synthetic Division!!! Sec. 3.3a Homework: p. 373-374 1-31 odd

We use a similar process when dividing polynomials!!! First, let’s work through this… We can write our answer as: We use a similar process when dividing polynomials!!! Remainder!

Division Algorithm for Polynomials Let f(x) and d(x) be polynomials with the degree of f greater than or equal to the degree of d, and d(x) = 0. Then there are unique polynomials q(x) and r(x), called the quotient and remainder, such that where either r(x) = 0 or the degree of r is less than the degree of d. The function f(x) is the dividend, d(x) is the divisor, and if r(x) = 0, we say d(x) divides evenly into f(x). Fraction form:

Using Polynomial Long Division Use long division to find the quotient and remainder when is divided by . Write a summary statement in both polynomial and fraction form.  Quotient  Remainder

Can we verify these answers graphically??? Using Polynomial Long Division Use long division to find the quotient and remainder when is divided by . Write a summary statement in both polynomial and fraction form. Polynomial Form: Fraction Form: Can we verify these answers graphically???

Synthetic Division Synthetic Division is a shortcut method for the division of a polynomial by a linear divisor, x – k. Notes: This technique works only when dividing by a linear polynomial… It is essentially a “collapsed” version of the long division we practiced last class…

Synthetic Division – Examples: Evaluate the quotient: Coefficients of dividend: Zero of divisor: 3 2 –3 –5 –12 6 9 12 2 3 4 Remainder Quotient

1 –2 3 –3 –2 4 –4 2 1 –2 2 –1 –1 Verify Graphically? Synthetic Division – Examples: Divide by and write a summary statement in fraction form. –2 1 –2 3 –3 –2 4 –4 2 1 –2 2 –1 –1 Verify Graphically?

Practice Divide the above function by Divide the above function by

Yes, x – 3 is a factor of the second Using Our New Theorems Is the first polynomial and factor of the second? Yes, x – 3 is a factor of the second polynomial

Some whiteboard problems…

“Fundamental Connections” for Polynomial Functions For a polynomial function f and a real number k, the following statements 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).