The Factor Theorem. Using The Factor Theorem Method To Prove That A Polynomial Has A Root.


The factor theorem can be used to locate the roots of a polynomial. The roots are where the polynomial crosses the x-axis. The factor theorem states that:

If (x- a) is a factor of f(x) then f(x) = 0 or if f(ax-b) is a factor of f(x) the f(b/a) = 0

Let’s put this factor theorem into practice:

Example 1

Show that x-3 is a factor of f(x) = x² - x – 6.

First make x -3 = 0 and solve this equation to give x = 3.

So all you need to do now is work out the value of f(3).

f(3) = 3² - 3 -6 = 9 – 3 -6 = 0

So by the factor theorem then x – 3 is a factor of f(x)

Example 2

Show that x+2 is a factor of f(x) = x² - 3x – 10.

First make x + 2 = 0 and solve this equation to give x = -2.

So all you need to do now is work out the value of f(-2).

f(3) = (-2)² - 3×(-2) -10 = 4 + 6 – 10 = 0

So by the factor theorem then x + 2 is a factor of f(x).

Example 3

Show that x-7 is a factor of f(x) = x² - 8x +7.

First make x - 7 = 0 and solve this equation to give x = 7.

So all you need to do now is work out the value of f(7).

f(7) = (7)² - 8×(7) +7 = 49 – 56 + 7 = 0

So by the factor theorem then x - 7 is a factor of f(x)

The next two examples are a little bit harder

Example 4

Show that 3x-2 is a factor of f(x) = 3x² + 10x – 8.

First make 3x - 2 = 0 and solve this equation to give x = 2/3

So all you need to do now is substitute f(2/3):

f(2/3) = 3(2/3)² + 10×(2/3) -8 = 12/9 + 20/3 – 8 = 12/9 + 60/9 – 72/9 = 0

So by the factor theorem then 3x - 2 is a factor of f(x).

Example 5

Show that x – 1 is a factor of f(x) = x³ + 2x² -x -2 = 0

First make x – 1 = 0, so the value of x we need is x = 1.

So substitute f(1) into this cubic equation:

f(1) = 1³ + 2×1² - 1 -2 = 1 + 2 -1 -2 = 0.

Therefore by the factor theorem, x -1 is a factor of f(x).

So as you can see the factor theorem is quite easy to carry out. All you need to do is set the factor equal to 0, solve this equation and then substitute this value into f(x) which should give you 0.

The factor theorem can be used to help you factorise fully harder polynomials.

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