SOLUTION: which of the following polynomials has a factor of x-1?. A) {{{p(x)=x^3+x^2-2x+1}}} B) {{{q(x)=2x^3-x^2+x-1}}} c) {{{r(x)=3x^3-x-2}}} D) {{{s(x)=-3x^3+3x+1}}}

Question 1040787: which of the following polynomials has a factor of x-1?.
A)
B)
c)
D)
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
By the remainder theorem, if x-k is factor of p(x), then p(k) = 0.

In this case, x-k = x-1, so k = 1. Therefore, if we find a function p(x) such that p(k) = p(1) = 0 then we have found the answer.

Essentially we plug in x = 1 into each answer choice. Then we see which results in 0.

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Let's go through the answer choices one by one. Starting with choice A

Replace every x with 1

Since the result is NOT equal to 0, this means that x-1 is NOT a factor of p(x).

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Now onto choice B. Like before, plug in x = 1 to see if we get a result of 0.

Replace every 'x' with 1

Again we don't get 0. So choice B is not the answer either.
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Part C)

Plug in x = 1

We got a result of 0, so x-1 is a factor of r(x). We have our answer. Let's check part D just to complete the problem.
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Part D)

Plug in x = 1

The nonzero result means x-1 is not a factor of s(x)
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To summarize, the final answer is Choice C
Answer by ikleyn(52857) (Show Source): You can put this solution on YOUR website!
.
which of the following polynomials has a factor of x-1?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Those and only those that have the number "1" as their root.

(based on the "Remainder theorem", see the lesson Divisibility of polynomial f(x) by binomial x-a in this site).

A) = B) = c) = <--- This and only this D) =

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