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The Factor Theorem. The Factor Theorem. Suppose that a polynomial is divided by an expression of the form ( x – a ) the remainder is 0. What can you conclude about ( x – a )?. If the remainder is 0 then ( x – a ) is a factor of the polynomial. This is the Factor Theorem :.

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the factor theorem
The Factor Theorem

Suppose that a polynomial is divided by an expression of the form (x – a) the remainder is 0.

What can you conclude about (x – a)?

If the remainder is 0 then (x – a) is a factor of the polynomial.

This is the Factor Theorem:

If (x– a) is a factor of a polynomial then substituting

a in x will give an answer of zero.

The converse is also true:

If substituting a in for x gives an answer of zero

then (x– a) is a factor of a polynomial.

the factor theorem1
The Factor Theorem

And so

2x2 - 7x + 3 = (x - 3)(2x – 1)

Use the Factor Theorem to show that

(x - 3) is a factor of 2x2 - 7x + 3.

(x - 3) is a factor of 2x2 - 7x + 3 if we substitute 3 in

for x and get zero.

2x2 - 7x + 3 = 2(3)2 - 7(3) + 3

= 18 – 21 + 3

= 0 as required.

the factor theorem2
The Factor Theorem

And so

3x2 + 5x – 2 = (x + 2)(3x – 1)

Use the Factor Theorem to show that

(x + 2) is a factor of 3x2 + 5x – 2.

(x + 2) is a factor of 3x2 + 5x – 2 if we substitute –2 in

for x and get zero.

3x2 + 5x – 2 = 3(–2)2 + 5(–2) – 2

= 12 – 10 – 2

= 0 as required.

factoring polynomials
Factoring polynomials

The Factor Theorem can be used to factor polynomials by systematically looking for values of x that will make the polynomial equal to 0. For example:

Factor the cubic polynomial x3 – 3x2 – 6x + 8.

Let x3 – 3x2 – 6x + 8.

Start by testing 1, then -1, then 2, etc.

x3 – 3x2 – 6x + 8 = 1 – 3 – 6 + 8 = 0

 (x – 1) is a factor of x3 – 3x2 – 6x + 8.

Now long divide