Factor Theorem

AN EXAMPLE

You wish to find the factors of
:

x^3 + 7x^2 + 8x + 2

To do this you would use trial and error finding the first factor. When the result is equal to

0

, we know that we have a factor. Is

(x - 1)

a factor? To find out, subtitute

x = 1

into the polynomial above:
:

(1^3) + 7(1^2) + 8(1) + 2

This is equal to

18

not

0

(x - 1)

is not a factor of

x^3 + 7x^2 + 8x + 2

. So, we next try

(x + 1)

(substituting

x = -1

into the polynomial):
:

(-1^3) + 7(-1^2) + 8(-1) + 2

This is equal to

0

. Therefore

x-(-1)

(or rather

x+1

) is a factor, and -1 is a Root of

x^3 + 7x^2 + 8x + 2

The next two roots can be found by Algebraicly Dividing

x^3 + 7x^2 + 8x + 2

(x+1)

to get a quadratic, which can be solved directly, by the factor theorem or by the Quadratic Equation .

(x^3 + 7x^2 + 8x + 2) \over (x + 1)

x^2 + 6x + 2

and therefore

(x+1)

and

x^2 + 6x + 2

are the factors of

x^3 + 7x^2 + 8x + 2

FORMAL VERSION

More formally, it states that for any polynomial
:

f(x)

,

for all values of

a

which satisfy
:

f(a) = 0

,

(in which the value of a is substituted for x into the "y=" equation)

(x - a)

is a Factor of

f(x)

. Or, more concisely:

:

rac{f(x)}{x-a} = q(x)

is a polynomial.

This indicates that any a for which f(-a) = 0, is a root of f(x). Double roots can be found by performing polynomial long division.

	CATEGORIES ABOUT FACTOR THEOREM

	polynomials

	mathematical theorems

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