Use synthetic division to find the quotient and remainder

Question

Use synthetic division to find the quotient and remainder when

f(x)= 9x^4+2x^3+1x^2+9x-2 is divided by

g(x)= x-2

the quotient is?

the remainder is?

please show step by step 

Answer 1

When using synthetic division it is important to make sure the expression that you are dividing is arranged from left to right starting with the highest power (exponent) to the lowest. Also make sure there are no powers missing, for example  X⁴ + 3X² - X¹ + 9, notice there is no X³. If a power is missing, represent it with a coefficient of zero such as X⁴ + 0X³  +3X²- X¹ + 9

 

Once you have made sure your expression is set up correctly you need to determine what number you are dividing by, in the example you posted, you are dividing by the factor X-2 so you would use 2. Some teachers say use the opposite of what is shown so if it were X+3 I would use -3.

 

When you use synthetic division, you only need the coefficients. So lets look at your problem.

 

9X⁴ + 2X³+X²+9X¹ - 2  ÷ X-2   remember if there is no number (coefficient) in front of the X you use a 1

 

This gives    2⌋  9      2      1      9      -2

                        ↓     18    40    82    182

                        9    20    41    91    180

 

Step 1 is to bring the first number down under the line.

 

Step 2 multiply the number out front, (in this problem a 2) by the 9 and place that number under the second

           number. Add the two numbers.

 

Step 3 keep repeating the process in step 2 for each number under the line until you run out of numbers. The

          number left under the line to the far right is your remainder.

 

Step 4 use the numbers under the line as the coefficients for your new expressions and always began with a

          power one less than the original expression. So for this equation you would end up with

          9X³ + 20X³+41X²+91X +180  this is your quotient and your remainder was 180.

 

 

Answer 2

            9X^3 + 20 X^2 + 41X + 91                                    

   X - 2 l 9 X^4 + 2 X^3 + X^2 + 9X - 2

             9X^4  ±18X^3           

                       20 X^3  + X^2

                       20 X^3  ±42X^2    

                                    41X^2+9X

                                    41X^2±82x

                                               91x - 2

                                               91X±182

                                                      180

 

 Quotient : 9X^3 + 20 X^2 + 43X + 91

 Remainder : 180

 

 

  The step by step procedure is like do any division:

 

       _25_____  

    7 l 176

       -14    

         36

         -35   

           1

 

     176 - dividend                7 -- Divisor        25- Quotient       1 - remainder