How do you use synthetic division to find the factors of #f(x)= x^4 -x^3 -19x^2+49x-30#?

1 Answer
Aug 30, 2015

Answer:

If you can spot a factor of #f(x)# then you can use synthetic division to divide by that factor to give a simpler polynomial to factor.

Hence we can find:

#f(x) = (x-1)(x+5)(x-2)(x-3)#

Explanation:

First note that #f(1) = 1 - 1 - 19 + 49 - 30 = 0#, so #(x - 1)# is a factor of #f(x)#

Use synthetic division to divide #f(x)# by #(x-1)#:
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So #f(x) = (x-1)(x^3-19x+30)#

Notice that #(-5)^3-19*(-5)+30 = -125+95+30 = 0#, so #(x+5)# is also a factor of #f(x)#.

Divide #x^3-19x+30# by #(x+5)# using synthetic division - not forgetting to specify the coefficient #0# of the #x^2# term:
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So #f(x) = (x-1)(x+5)(x^2-5+6)#

By this stage you can probably spot that #x^2-5+6 = (x-2)(x-3)# to complete our factorisation:

#f(x) = (x-1)(x+5)(x-2)(x-3)#