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Precalculus: Complete Set of Lessons

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P01-LongDivisionPolys.mws

High School Modules > Precalculus by Gregory A. Moore

Long Division of Polynomials


Division of polynomials by binomials of degree ones using long division.

[Directions : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in this worksheet.]

0. Code

> restart;

> libname:="C:\\Program Files\\Maple 7\\LIB\\Precalc\\", libname:

1. Long Division of Polynomials


You are no doubt familiar with the method of converting an improper fraction to a mixed number by performing a long division.

> 17/3;

17/3

> MixedNumber(17,3);

17/3 = 5, ` + `, 2/3

> MixedNumber( 40,13);

40/13 = 3, ` + `, 1/13

> MixedNumber( 177,19 );

177/19 = 9, ` + `, 6/19

> MixedNumber( 4001, 99 );

4001/99 = 40, ` + `, 41/99


Just as we divide number using long division, we can also divide polynomials.

Here are examples of this kind of division - without showing the steps of the method yet.

> PolyDivide( x^2 + 1, x - 1);

(x^2+1)/(x-1) = 1+x+2/(x-1)

> PolyDivide(x^3 + 4*x + 17 ,x-3 );

(x^3+4*x+17)/(x-3) = x^2+3*x+13+56/(x-3)

> PolyDivide( 3*x^5-x^4-29*x^3-27*x^2-114*x-72, x - 3);

(3*x^5-x^4-29*x^3-27*x^2-114*x-72)/(x-3) = 3*x^4+8*...

> PolyDivide( x^7 - 1, x - 3);

(x^7-1)/(x-3) = x^6+3*x^5+9*x^4+27*x^3+81*x^2+243*x...


Just as with numbers, sometimes we get lucky, and the denomenator goes into the numerator exactly and their is no remainder.

> PolyDivide( x^5 + 1, x + 1);

(x^5+1)/(1+x) = x^4-x^3+x^2-x+1

> PolyDivide( 3*x^5-x^4-29*x^3-27*x^2-114*x-72, x + 3);

(3*x^5-x^4-29*x^3-27*x^2-114*x-72)/(x+3) = 3*x^4-10...

> PolyDivide( 3*x^5-x^4-29*x^3-27*x^2-114*x-72, x - 4);

(3*x^5-x^4-29*x^3-27*x^2-114*x-72)/(x-4) = 3*x^4+11...

> PolyDivide( 256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4
+ 108864*x^3 + 81648*x^2 + 34992*x + 6561,
2*x +3 );

(256*x^8+3072*x^7+16128*x^6+48384*x^5+90720*x^4+108...
(256*x^8+3072*x^7+16128*x^6+48384*x^5+90720*x^4+108...


Next we'll look at the actual method to compute these quotients and remainders.

2. The Method of Long Division


The method of long dividing polynomials has the same four steps as does numerical division :
1. DIVIDE
2. MULTIPLY
3. SUBTRACT
4. BRING DOWN


These four steps (DMSB = "Door Mouse Stores Bread" ) are performed repeatedly until there is nothing more to bring down. Lets look at an example.

> (x^2 + 5*x + 7) /(x+3);

(x^2+5*x+7)/(x+3)


We first set up a long division ...

> LongDiv1( x^2 + 5*x + 7, -3 );

matrix([[` `, ` `, ` `, ` `, ` `, ` `], [` `, ` `, ...


Now we go through the DMSB steps ....
1. DIVIDE ........... x into x^2 to get x
2. MULTIPLY
........... x by the divisor, x + 3 to get x^2 + 3x
3. SUBTRACT
........... the new product x^2 + 3x from x^2 + 5x to get 2x
4. BRING DOWN ........... 7 from the top row

> LongDiv2( x^2 + 5*x + 7, -3 );

matrix([[` `, ` `, ` `, ` `, ` `], [` `, ` `, __, _...




Now we go through the DMSB steps AGAIN
1. DIVIDE ........... x into 2x to get 2
2. MULTIPLY
........... 2 by the divisor, x + 3 to get 2x + 6
3. SUBTRACT ........... the new product 2x + 6 from 2x + 7 on the top row to get 1
4. BRING DOWN ........... (nothing more to bring down ... this means we are finished.....)

> LongDiv( x^2 + 5*x + 7, -3);

matrix([[` `, ` `, x, 2, ` `], [` `, ` `, __, __, _...

` `

` `

(x^2+5*x+7)/(x-3) = x+2+1/(x-3)

` `



Here are other examples. Try them by hand and verify that you get the same results.

>

> LongDiv( x^2 + 4*x + 10, 2);

matrix([[` `, ` `, x, 6, ` `], [` `, ` `, __, __, _...

` `

` `

(x^2+4*x+10)/(x+2) = x+6+22/(x+2)

` `

> LongDiv( x^2 + 1*x + 2, -2);

matrix([[` `, ` `, x, -1, ` `], [` `, ` `, __, __, ...

` `

` `

(x^2+x+2)/(x-2) = x-1+4/(x-2)

` `

> LongDiv( x^3 + 6*x^2 + 8*x + 2, -3);

matrix([[` `, ` `, x^2, 3*x, -1, ` `], [` `, ` `, _...

` `

` `

(x^3+6*x^2+8*x+2)/(x-3) = x^2+3*x-1+5/(x-3)

` `



You can also have fractions in the divisor.

> LongDiv( 6*x^2 + 8*x + 2, -1/3);

matrix([[` `, ` `, 6*x, 6, ` `], [` `, ` `, __, __,...

` `

` `

(2+6*x^2+8*x)/(x-1/3) = 6*x+6

` `


Whenever there are missing terms, its necessary to put a "placeholder" - a zero term - in its place, so that everything works smoothly.

> LongDiv( x^2 + 100, -5);

matrix([[` `, ` `, x, -5, ` `], [` `, ` `, __, __, ...

` `

` `

(x^2+100)/(x-5) = x-5+125/(x-5)

` `

> LongDiv( x^3 + 5*x + 7, -2);

matrix([[` `, ` `, x^2, -2*x, 9, ` `], [` `, ` `, _...

` `

` `

(x^3+5*x+7)/(x-2) = x^2-2*x+9-11/(x-2)

` `

> LongDiv( x^4 + 9*x^4 + 7, -1);

matrix([[` `, ` `, 10*x^3, -10*x^2, 10*x, -10, ` `]...

` `

` `

(10*x^4+7)/(x-1) = 10*x^3-10*x^2+10*x-10+17/(x-1)

` `


The most common places that people make mistakes are when some of the numbers are negative. We must remember to always subtract no matter whether the numbers are positive or negative or mixed.

> LongDiv( x^2 - 5*x -2, 4);

matrix([[` `, ` `, x, -1, ` `], [` `, ` `, __, __, ...

` `

` `

(x^2-5*x-2)/(x+4) = x-1-6/(x+4)

` `

> LongDiv( x^2 + 3*x -2, 7);

matrix([[` `, ` `, x, 10, ` `], [` `, ` `, __, __, ...

` `

` `

(x^2+3*x-2)/(x+7) = x+10+68/(x+7)

` `


3. Dividing Larger Polynomials


Above,we divided polynomials by polynomials of degree 1 with leading coefficient 1. We can also divide by larger polynomials using the same method. You can do these by hand, and check in Maple.

> PolyDivide( x^3 + 11*x^2 - 4*x + 17, 3*x +5);

(x^3+11*x^2-4*x+17)/(3*x+5) = 1/3*x^2+28/9*x-176/27...

> PolyDivide( 2000*x^3 + 5, 10*x - 9);

(2000*x^3+5)/(10*x-9) = 200*x^2+180*x+162+1463/(10*...

> PolyDivide( x^4 - 1, x^2 + 1);

(x^4-1)/(x^2+1) = x^2-1

> PolyDivide( x^4 + 1, x^2 + 1);

(x^4+1)/(x^2+1) = x^2-1+2/(x^2+1)

> PolyDivide( x^4 + 2*x^3 + 4*x^2 + 6*x + 8, x^2 + 2*x + 4);

(x^4+2*x^3+4*x^2+6*x+8)/(x^2+2*x+4) = x^2+(8+6*x)/(...

4. Factoring Polynomials


When you divide polynomials and there is no remainder, this means that one polynomial divides the other. Thus the numerator can be factored.

> PolyDivide( 12*x^2-155*x-375 , x - 15);

(12*x^2-155*x-375)/(x-15) = 12*x+25

> 12*x^2-155*x-375 = (x - 15)*(12*x + 25);

12*x^2-155*x-375 = (x-15)*(12*x+25)


Write out the factorization for each example.

> PolyDivide( 15*x^3+53*x^2-69*x+400, 3*x + 16);

(15*x^3+53*x^2-69*x+400)/(3*x+16) = 5*x^2-9*x+25

> PolyDivide( x^4 + 2*x^3 + 4*x^2 + 6*x + 8, x^2 + 2*x + 4);

(x^4+2*x^3+4*x^2+6*x+8)/(x^2+2*x+4) = x^2+(8+6*x)/(...

> PolyDivide( 168*x^3+731*x^2-862*x+96, 7*x - 6 );

(168*x^3+731*x^2-862*x+96)/(7*x-6) = 24*x^2+125*x-1...

> PolyDivide( 6*x^4-19*x^3-106*x^2+21*x+18, x+3 );

(6*x^4-19*x^3-106*x^2+21*x+18)/(x+3) = 6*x^3-37*x^2...

> PolyDivide( 6*x^4-19*x^3-106*x^2+21*x+18, 3*x+1 );

(6*x^4-19*x^3-106*x^2+21*x+18)/(3*x+1) = 2*x^3-7*x^...


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