The Remainder and Factor Theorems

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The Remainder and Factor Theorems

• 6.5
• p. 352

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When you divide a Polynomial f(x) by a divisor
d(x), you get a quotient polynomial q(x) with a
remainder r(x) writtenf(x) q(x) r(x)d(x)
d(x)
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The degree of the remainder must be less than the
degree of the divisor!
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Polynomial Long Division

• You write the division problem in the same format
  you would use for numbers. If a term is missing
  in standard form fill it in with a 0
  coefficient.
• Example
• 2x4 3x3 5x 1
• x2 2x 2

5
2x2
2x4 2x2 x2
6
2x2
7x
10
-( )
4x2
-4x3
2x4
- 4x2
7x3
5x
7x3 - 14x2 14x
-( )
10x2 - 9x
-1
7x3 7x x2
10x2 - 20x 20
-( )
11x - 21
remainder
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The answer is written

• 2x2 7x 10 11x 21 x2 2x 2
• Quotient Remainder over divisor

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Now you try one!

• y4 2y2 y 5 y2 y 1
• Answer y2 y 2 3
  y2 y 1

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Remainder Theorem

• If a polynomial f(x) is divisible by (x k),
  then the remainder is r f(k).
• Now you will use synthetic division (like
  synthetic substitution)
• f(x) 3x3 2x2 2x 5
• Divide by x - 2

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f(x) 3x3 2x2 2x 5 Divide by x - 2

• Long division results in ?......
• 3x2 4x 10 15 x 2
• Synthetic Division
• f(2) 3 -2 2 -5 2

6
8
20
3
4
10
15
Which gives you
15 x-2
3x2
10
4x
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Synthetic Division Practice 1

• Divide x3 2x2 6x -9 by (a) x-2 (b) x3
• (a) x-2
• 1 2 -6 -9 2

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4
2
1
4
2
-5
Which is x2 4x 2 -5 x-2
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Synthetic Division Practice cont.

• (b) x3
• 1 2 -6 -9 -3

3
9
-3
1
-1
-3
0
x2 x - 3
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Factor Theorem

• A polynomial f(x) has factor x-k iff f(k)0
• note that k is a ZERO of the function because
  f(k)0

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Factoring a polynomial

• Factor f(x) 2x3 11x2 18x 9
• Given f(-3)0
• Since f(-3)0
• x-(-3) or x3 is a factor
• So use synthetic division to find the others!!

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Factoring a polynomial cont.

• 2 11 18 9
• -3

-15
-9
-6
2
5
3
0
So. 2x3 11x2 18x 9 factors to
(x 3)(2x2 5x 3)
Now keep factoring (bustin da b) gives you
(x3)(2x3)(x1)
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Your turn!

• Factor f(x) 3x3 13x2 2x -8
• given f(-4)0
• (x 1)(3x 2)(x 4)

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Finding the zeros of a polynomial function

• f(x) x3 2x2 9x 18.
• One zero of f(x) is x2
• Find the others!
• Use synthetic div. to reduce the degree of the
  polynomial function and factor completely.
• (x-2)(x2-9) (x-2)(x3)(x-3)
• Therefore, the zeros are x2,3,-3!!!

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Your turn!

• f(x) x3 6x2 3x -10
• X-5 is one zero, find the others!
• The zeros are x2,-1,-5
• Because the factors are (x-2)(x1)(x5)

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Assignment