Presentation on theme: "Long Division Algorithm and Synthetic Division!!!"— Presentation transcript:
1 Long Division Algorithm and Synthetic Division!!! Sec. 3.3aHomework: p odd
2 We use a similar process when dividing polynomials!!! First, let’s work through this…We can write our answer as:We use a similar processwhen dividing polynomials!!!Remainder!
3 Division Algorithm for Polynomials Let 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 thatwhere 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:
4 Using Polynomial Long Division Use long division to find the quotient and remainder whenis divided by Write a summarystatement in both polynomial and fraction form. Quotient Remainder
5 Can we verify these answers graphically??? Using Polynomial Long DivisionUse long division to find the quotient and remainder whenis divided by Write a summarystatement in both polynomial and fraction form.Polynomial Form:Fraction Form:Can we verify these answers graphically???
6 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 alinear polynomial…It is essentially a “collapsed” version of the longdivision we practiced last class…
7 Synthetic Division – Examples: Evaluate the quotient:Coefficients of dividend:Zero ofdivisor:32–3–5–126912234RemainderQuotient
8 1 –2 3 –3 –2 4 –4 2 1 –2 2 –1 –1 Verify Graphically? Synthetic Division – Examples:Divide by and write asummary statement in fraction form.–21–23–3–24–421–22–1–1Verify Graphically?
9 PracticeDivide the above function byDivide the above function by
10 Yes, x – 3 is a factor of the second Using Our New TheoremsIs the first polynomial and factor of the second?Yes, x – 3 is a factor of the secondpolynomial
12 “Fundamental Connections” for Polynomial 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).