Finding Values of a Polynomial . show f(x) in factorised form..is this correct
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Finding Values of a Polynomial . show f(x) in factorised form..is this correct

[From: ] [author: ] [Date: 12-08-09] [Hit: ]
-5...-4.........

a = -40/8 = -5 ---> b = -2 - a = -2 - (-5) = 3

a = -5, b = 3

f(x) = 2x^3 -5x^2 - 4x+3

you have to find the 3rd factor Q(x) = cx + d (f(x) is 3rd order and (x+1)(x-3) is 2nd order, so Q(x) must be 1st order for the multiplication of the three terms to give something with a cubic highest term

apply horner's synthetic division:

-1 || 2...-5...-4....3
--------------------------------
..........-2....7...-3
--------------------------------
......2...-7....3....0

2x^3 -5x^2 - 4x+3 = (x+1)(2x^2-7x+3)

then do it again for x = 3

3||....2...-7....3
----------------------------
.............6....-3
----------------------------
.......2....-1....0

(2x^2-7x+3) = (x-3)(2x-1)

so

f(x) = (x+1)(x-3)(2x-1)


Short Cut
----------------
You can use this method for all degree equations as it's a contracted form of long division. But you can use a short cut when you know two factors out of three like in this question:

(x+1)(x-3) = x*x+....+(1)(-3)

likewise

2x^3 -5x^2 - 4x+3 = (x+1)(x-3)(cx+d)

x*x*(cx) = 2x^3

cx^3 = 2x^3 ---> c=2, and

(1)*(-3)*(d) = 3 ----> -3d = 3 ---> d= -1, so

cx + d = 2x -1

you can get the last term this way too.

(didn't realize my horners were out of shape so had to add dots to get them in line)
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