How do you determine whether x-1 is a factor of the polynomial 3x^3-x-33x3x3?

1 Answer
Jan 19, 2017

(x-1)(x1) is not a factor of 3x^3-x-33x3x3

Explanation:

In general, if f(x)f(x) is a polynomial, then (x-a)(xa) is a factor if and only if f(a) = 0f(a)=0.

So in our example, we find:

f(color(blue)(1)) = 3(color(blue)(1)^3)-color(blue)(1)-3 = 3-1-3 = -1 != 0f(1)=3(13)13=313=10

So (x-1)(x1) is not a factor of 3x^3-x-33x3x3

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Footnote

If you want to quickly check whether x=1x=1 is a zero and therefore (x-1)(x1) is a factor of a polynomial, just add the coefficients and see if the result is 00.

This is the same as evaluating the polynomial for x=1x=1 since 11 raised to any integer power is 11.

You can also check whether x=-1x=1 is a zero and (x+1)(x+1) a factor by reversing the signs of the coefficients of the terms of odd degree before adding them.