The Factor Theorem - By David Spollen
How to use this applet:The Factor Theorem:
-Enter the co-efficients (a, b, c, d) of any given
cubic polynomial in the boxes provided for them.
-Enter the value of x in its box provided.
-Click on the "Enter x" button.
-An equation should appear containing your values
and a report as to whether your x expression is a factor of your polynomial or not.Notes on the maths used in the applet:The Remainder Theorem states that when a polynomial f(x) is divided by (x-a), the remainder is f(a). Therefore, if (x-a) is a factor of f(x), the remainder will be zero and
f(x) = 0. Also, for any k ª R, (x-k) is a factor of f(x) - f(k).
Proof of the factor theorem:
In any cubic polynomial,
f(x) = ax^3 + bx^2 + cx + d
hence
f(k) = ak^3 + bk^2 + ck + d
f(x) – f(k) = (ax^3 + bx^2 + cx + d ) – (ak^3 + bk^2 + ck + d)
= a(x^3 - k^3) + b(x^2 - k^2)
+ c(x –k)
= a(x – k)(x^2 + xk +k^2) b(x – k)(x + k) + (c – k)
= (x – k)[a(x^2 + xk + k^2) +
b(x + k) + c]
But f(k) = 0, therefore (x – k) is a factor of
f(x)
In general, if (i) (x – a) is a factor of f(x), then f(a) = 0.
(ii) (ax – b) is a factor of f(x), then f(b/a) = 0.
Example
Q: If (2x - 1) is a factor of 2x^3-5x^2-kx+3 find the value of k and hence the two other factors.
A: f(1/2) = 0
f(1/2) = 2(1/2)^3 - 5(1/2)^2 - kx + 3 = 0
=> k = 4
f(x) = 2x^3 - 5x^2 - 4x + 3
Dividing (2x - 1) into f(x) we get a remainder of 0 and the quadratic x^2- 2x - 3. Therefore the other two factors are (x - 3) and (x + 1).
Some other key pointers:
-If f(x) = ax^3 + bx^2 + cx + d and if f(1) = 0,
then (x - 1) is a factor of f(x). But if f(1) is not equal to 0, we then proceed to check if
f(-1) = 0 and so on with f(2), f(-2), f(3), f(-3)
etc. until we find a root equal to zero to give us the factor.
-If (x - t) is a factor of ax^3 + bx^2 + cx + d
then t has to be a factor of d.
-If we have no x term in a given polynomial in a question, we must insert a 0x term. The same applies for x^3 or x^2 terms.
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