Synthetic Division/Rational Roots

-- Section 5.3 --

 
Use synthetic division to determine if (x - 1) is a factor of  P(x) = x3 - 2x2 - 5x + 6
Put 1 in the upper left hand corner (box). Put the coefficients of the polynomial in the first row. 

Solve

            Step

  

  
Bring the down the 1. Use  1 (in box) if  x - 1 is the factor.
For x
- r , use r


  

Multiply 1 1 = 1

Add Columns -2 + 1 = -1


  

Multiply -1 1 = 1

Add Columns -5 + (-1) = -6

 
  

Multiply -6 1 = -6

Add Columns 6 + (-6) = 0

If there is a remainder of zero, 0 , then 1 is a zero and (x -1) is a factor of the polynomial.

  
Use synthetic division to determine if (x - 1) is a factor of  P(x) = 2x3 - 2x2 - 5x + 6

[Solution]

Find all rational roots of  P(x) = 2x3 + 3x2 - 14x - 15
 
Find all possible rational zeros - rational numbers are ratio of integers i.e. -2/1 = -2, 0, 1/2, -3/2

Solve

            Step

   
p = ± 1,  ± 3,  ± 5,  ± 15

q ± 1, ± 2

Find possible rational zeros
p/q where p are factors of the last coefficient 15.  q are the factors of the first coefficient 2.

p = ± 1,  ± 2,  ± 3,  ± 6  and 

p/1 = p  and all of the p/2's. We always get the p's

Try the easy ones first.
Start with -1 (I tried 1 first but it failed - You can use Descartes' Rules of Signs to narrow the search.)
Put -1 in the upper left hand corner (box). Put the coefficients of the polynomial in the first row.

  

Bring the down the 2. coefficient of 2x3 is 2

 

Multiply 2 -1 = -2

Add Columns 3 + (-2) = 1

Multiply 1 -1 = -1

Add Columns -14 + (-1) = -15

     Multiply -15 -1 = 15

Add Columns -15 + 15 = 0

If there is a remainder of zero, 0 , then -1 is a zero and (x -(-1)) = (x + 1) is a factor of the polynomial.

 Q(x) = 2x2 + x - 15

The quotient is the coefficients of the bottom line Once we have a quadratic equation, factor or use the quadratic formula.

2x2 + x - 15 = (2x - 1)(x + 3) = 0

Set equal to zero and solve for x

2x - 1 = 0   or   x + 3 = 0

x = 1/2   or x = -3

Check in original equation.

{ -1, 1/2, -3}

There are three solutions for P(x) P(1/2) = P(-1) = P(-3) = 0

Side note:  P(x) = 2x3 + 3x2 - 14x - 15 = (x + 1)(2x - 1)(x + 3)

 
Find all possible rational roots and all rational roots of P(x) = 4x3 - 15x2 - 31x + 30

[Solution]

Find all roots of  P(x) = 3x4 + 10x3 - 9x2 - 40x - 12
Find all possible rational zeros - rational numbers are ratio of integers i.e. -2/1 = -2, 0, 1/2, -3/2

Solve

            Step

    Property
p = ± 1,  ± 2,  ± 3,  ± 4, ± 6,  ± 12

q ± 1, ± 3

Find possible rational zeros
p/q where p are factors of the last coefficient 12.  q are the factors of the first coefficient 2.

p = ± 1,  ± 2,  ± 3,  ± 4, ± 6,  ± 12 and

p/1 = p   We always get the p's
and all of the p/3's.
Reduce
Start with 2 (I tried others first - You can use Descartes' Rules of Signs to narrow the search.)
Bring the down the 3. coefficient of 3x4 is 3
Multiply 3 2 = 6

Add Columns 10 + 6 = 16

Multiply 16 2 = 32

Add Columns -9 + 32 = 23

Multiply 23 2 = 46

Add Columns -40 + 46 =  6

Multiply 6 2 = 12

Add Columns -12 + 12 = 0

If there is a remainder of zero, 0 , then 2 is a zero and (x - 2)  is a factor of the polynomial.

 Q(x) = 3x3 + 16x2 + 23x + 6

The quotient is the coefficients of the bottom line
Try -2 next I tried other numbers that failed first
I did the work for you....
If there is a remainder of zero, 0 , then 2 is a zero and (x - (-2)) = (x + 2) is a factor of the polynomial.
Q(x) = 3x2 + 10x + 3 The quotient is the coefficients of the bottom line Once we have a quadratic equation, factor or use the quadratic formula.

3x2 + 10x + 35 = (3x + 1)(x + 3) = 0

Set equal to zero and solve for x

3x + 1 = 0   or   x + 3 = 0

x = -1/3   or x = -3

Check in original equation.

{ 2, -2, -1/3  , -3}

There are three solutions for P(x) P(2) = P(-2) = P(-1/3) = P(-3) = 0
Side note:   P(x) = 3x4 + 10x3 - 9x2 - 40x - 12 = (x - 2)(x + 2)(3x + 1)(x + 3)

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada