Dividing Polynomials 2002 by Shawna Haider Dividing by
Dividing Polynomials © 2002 by Shawna Haider
Dividing by a Monomial If the divisor only has one term, split the polynomial up into a fraction for each term. divisor Now reduce each fraction. 3 x 3 4 x 2 1 1
Long Division If the divisor has more than one term, perform long division. You do the same steps with polynomial division as with integers. Let's do two problems, one with integers you know how to do and one with polynomials and copy the steps. Subtract (which Now multiply by x + 11 21 changes the sign Bring down the divisor and put Multiply and 2 + 8 x - 5 32 698 x 3 x of each term in next number or the answer below. put below Remainder 2 – 3 x 64 x the polynomial) term subtract added here 58 3 into 6 First divide or x into x 2 11 x - 5 over divisor 32 Now divide is the x into 11 x - 33 26 3 into 5 This or remainder 28 So we found the answer to the problem x 2 + 8 x – 5 x – 3 or the problem written another way:
Let's Try Another One If any powers of terms are missing you should write them in with zeros in front to keep all of your columns straight. Subtract (which y-2 Write out with changes theand sign Multiply Bring Multiply down and the 2 + 0 y + 8 long division 2 y + 2 y Divide intoiny-2 y Divide yyinto of each term put below next put term below including 0 y for Remainder 2 + 2 y y the polynomial) subtract missing term added here This is the remainder -2 y + 8 over divisor - 2 y - 4 12
Synthetic Division There is a shortcut for long division as long as the divisor is x – k where k is some number. (Can't have any powers on x). Set divisor = 0 and 1 solve. Put answer here. x + 3 = 0 so x = - 3 -3 1 6 8 -2 up 3 these Bring number down below Addupthese line up - 3 first. Add - 9 these. Add Multiply these and these 2 +3 x - 1 This is the remainder 1 x 1 put answer above line above Put variables back in (one was of divided outthe in Sonext the Listanswer all coefficients is: (numbers in xfront x's) and in in next process so first number is one less power than constant along the top. If a term is missing, put in a 0. column original problem).
Let's try another Synthetic Division Set divisor = 0 and solve. Put answer here. 4 1 0 x 3 0 x 1 x - 4 = 0 so x = 4 0 -4 0 6 up 48 Bring number down these below Add upthese line Add up these up 4 first. Add 16 these. Add 192 Multiply these and 3 + 4 x 2 + 12 x + 48 This is these and 1 x 198 put answer remainder put answer above line Now put variables back in was above lineanswer Sonext the List all coefficients is: (numbers in (remember front of x's)one andxthe in in next divided process first number is one less in next constant alongout theintop. Don'tsoforget the 0's for missing column 3 column terms. power than original problem so x ).
Let's try a problem where we factor the polynomial completely given one of its factors. -2 4 You want to divide the factor into the polynomial so set divisor = 0 and solve for first number. 8 -25 -50 up Bring number down below Addupthese line up - 8 first. Add 0 these. Add 50 these Multiply these and No remainder so x + 2 these 2 4 x + 0 x - 25 0 put IS a factor because it put answer above line divided in evenly above Put variables back in (one x was divided outthe in Sonext the Listanswer all coefficients is the divisor (numbers times in thefront quotient: of x's) and in in next process sothe first number is one less power You could check this by. If a term constant along top. is missing, putthan in a 0. column original problem). multiplying them out and getting original polynomial
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