Algebra 1 Synthetic Division
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How Unreal
Synthetic Division

 
 There is a shortcut that we can use
 to see if something like (X - 2) is a factor
 of some big polynomial like 2X 6 - 4X 5 + 3X 3 - 11X 2 + 17X - 14
 To use this shortcut, the factor that we're checking
 must start with X, not something like X 2 or 4X.
 
 Example:
 
 Check to see if (X - 2) is a factor of 
 

 2X6 - 4X5 + 3X3 - 11X2 + 17X - 14

 
 Here's what we do ...
 
 Step 1)
  Make sure that there is a term 
 for every power of X less than the biggest one.
 If any are missing (like X4 here) write a zero in it's place ...
 

 2X6 - 4X5 + 0 + 3X3 - 11X2 + 17X - 14

 
 Step 2)
 All powers of X are now present? Good!
 Now write down just the coefficients 
 (the numbers without the X's or exponents).
 

2 - 4 + 0 + 3 - 11 + 17 - 14

 
 Step 3)
  Now look at the term that we want to check.
 Look at it as:

 

 
 we need whatever goes in the box.

 Since our term is (X - 2), we have a 2 in the box.

 BUT if we were checking (X + 5) 
 we would need to write it as (X - (-5)).
 That means that -5 would be in the box.
 
 Step 4)  
 Take whatever goes in the box AND 
 the coefficients from step 2 and write them all down.
 Then draw a line under the numbers ...
 

 
 Step 5)
Take the first coefficient (here it's a 2)
 and write it below the line ...
 

 
 Step 6)
 Multiply the number that you wrote under the line
 times the number in the box.
 Write the answer under the next number to the right (2 x 2 = 4) ...
 

 
 Step 7)
 Add the number you just wrote down 
 and the number above it.
 Write the answer below the line (-4 + 4 = 0) ...
 

 
 Step 8)
 Repeat steps 6 and 7 until you run out of numbers on the right ...
 

 
 Step 9)
 IF the last number that you write down under the line is a zero,
 the thing that you were checking IS a factor of the big polynomial.
 If the last number is NOT a zero,
 the thing that you were checking is not a factor.
 
 Step 10) 
 If the thing that you were checking 
 is a factor of the big polynomial,
 the other factor is built from the numbers under the line.
 We write powers of X next to the numbers.
 Start on the left with an exponent one smaller
 than the biggest exponent in the big polynomial.
 As we go to the right, the exponents get one smaller each number.
 That is ...

2X5 + 0X4 + 0X3 + 3X2 - 5X + 7

 
 We can leave out any term with a zero coefficient.
 

2X5 + 3X2 - 5X + 7

 
 That means ...
 

 2X6 - 4X5 + 3X3 - 11X2 + 17X - 14 = (X - 2)(2X5 + 3X2 - 5X + 7)

 
 That's just swell.
 Now we've got good news, bad news, and good news.
 
 Good News: Once you get the hang of it,
 this is much easier to do than equation long division.
 
 Bad News: This trick only works with X (a number)
 and we still need to guess at what the factors might be.
 
 Good News: This little trick has another tricky use!
 
 Say that we wanted to find the value of:
 

 2X6 - 6X5 - 8X4 - X2 + 6X - 10    when X = 4

 
 Up to now, what we would have to do is calculate 4 6,
 then multiply that by 2, 
 then calculate 4 5,
 then multiply that by -6,
 add that to the first calculation, and so on.
 It's a big mess and takes forever!
 BUT NOW, we have a new way to figure this out.
 Here's what we do ...
 
 Set up the polynomial coefficients 
 just like we did in the example.
 Fill in any missing powers of X with zeros ...
 

2    -6    -8     0    -1     6    -10

 
 The X value that we want to solve for 
 goes in the box on the left ...
 

 
 Draw the line and do the trick ...
 

 
 Here's what makes this so great.
 The last number that you wrote down (the -2)
 is the value of 2X 6 - 6X 5 - 8X 4 - X 2 + 6X - 10 when X = 4.
 It's true! It works!
 Don't believe me?
 Check it out the old way ...
 

   copyright 2005 Bruce Kirkpatrick

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