Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided...

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Remainder and Factor Theorems

Transcript of Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided...

Page 1: Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look.

Remainder and Factor Theorems

Page 2: Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look.

REMAINDER THEOREMLet f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c).

Let’s look at an example to see how this theorem is useful.

1232 23 xxxxf

-2 2 -3 2 -1 using synthetic division let’s divide by x + 2

2 -7 16 -33

-4 14 -32

the remainderFind f(-2)

3312223222 23 f

So the remainder we get in synthetic division is the same as the answer we’d get if we put -2 in the function.

The root of x + 2 = 0 is x = -2

Page 3: Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look.

161835343 23 f

FACTOR THEOREM

Let f be a polynomial function. Then x – c is a factor of f (x) if and only if f (c) = 0

If and only if means this will be true either way:

1. If f(c) = 0, then x - c is a factor of f(x)

2. If x - c is a factor of f(x) then f(c) = 0.

12 -51 153-3 -4 5 0 8

-4 17 -51 161

?854 offactor a 3 Is 23 xxx

Try synthetic division and see if the remainder is 0

NO it’s not a factor. In fact, f(-3) = 161

We could have computed f(-3) at first to determine this. Not = 0 so not a factor

Opposite sign goes

here

Page 4: Remainder and Factor Theorems. REMAINDER THEOREM Let f be a polynomial function. If f (x) is divided by x – c, then the remainder is f (c). Let’s look.

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au

http://www.slcc.edu/
http://www.mathxtc.com/
http://www.ststephens.wa.edu.au/

Warm – up #2 Find the remainder when P(x) is divided by x – c.

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